Applications of integration
•Download as PPTX, PDF•
0 likes•2,021 views
The document discusses various methods for approximating the area under a curve including: - Left, right, and midpoint rectangular approximation methods which divide the area into rectangles to estimate it - Area methods involving integrating curves bounded by the x-axis, y-axis, or between two curves - Volume methods like the disk, washer, and shell methods that revolve curves around axes to find volumes and relate to areas It concludes that the rectangular approximation methods are used to estimate areas by dividing them into rectangles, and that the midpoint method provides the most accurate approximation.
Report
Share
Report
Share
Recommended
Welcome
This document discusses the areas of rectangles and triangles. It states that the area of a rectangle is equal to its length multiplied by its breadth. The area of a triangle is equal to one-half of the base multiplied by the height. It then demonstrates that if a line is drawn within a rectangle connecting one vertex to the opposite side, the area of the resulting triangle will be half the area of the original rectangle.
Area of quadrilaterals
This document discusses the area formulas for different types of quadrilaterals. It provides the formulas for calculating the area of rectangles, squares, parallelograms, rhombi, trapezoids, and general quadrilaterals. Examples are given to demonstrate how to use the formulas to calculate areas. The key defines the variables used in the formulas such as base, height, side, and diagonal lengths.
Math Investigation: Pythagorean Theorem
The document illustrates the Pythagorean theorem visually by showing how a larger square can be divided into four triangular regions and a smaller square that together make up the area of the largest square. It provides instructions for readers to cut out and arrange the regions themselves to see how the Pythagorean theorem works. It then suggests two investigation starting points: 1) drawing and cutting out a right triangle with leg lengths of 1 and 2 inches to cover the hypotenuse square, and 2) considering if the same is possible if the smaller leg square is subdivided instead.
017# areas of common shapes
This document discusses how to calculate the areas of common shapes like squares, rectangles, and parallelograms. It states that the area of a square is length times width or length times length, the area of a rectangle is length times width, and the area of a parallelogram is base times perpendicular height or can be calculated using the Pythagorean theorem.
Areas pedro
This document provides formulas for calculating the areas of various shapes including squares, triangles, rectangles, trapezoids, rhombuses, circles, semicircles, cylinders, and semicylinders. It states that the area of a square is side squared, a triangle is half base times height, a rectangle is base times height, a trapezoid is half the sum of the bases times the height, a rhombus can be half the product of the diagonals or base times height, the area of a circle is pi times the radius squared, a semicircle is pi times the radius squared over 2, and the area of a cylinder or semicylinder is pi times the radius squared times the height or half
Mensuration
The document provides formulas for calculating the perimeter and area of basic shapes like squares, rectangles, triangles, parallelograms, and circles. It defines each shape and gives the formula for finding their perimeter and area in terms of common properties like sides, width, length, base, height, radius, and diameter. For example, the perimeter of a square is defined as 4 times the length of one side, and the area of a square is the length of one side squared.
C4 EDEXCEL HELP
The document provides revision notes on various mathematics topics including:
1) Binomial expansions, partial fractions, trigonometry formulas, and techniques for integration like volumes of revolution.
2) Parametric equations, vectors, vector equations, planes, and differential equations.
3) Details are given for solving problems involving these topics, such as using compound angle formulas, rewriting algebraic fractions as partial fractions, and separating variables to solve first-order differential equations.
Mensuration
A MENSURATIO PPT WHICH GIVES EXACT KNOWLEDGE MUST SEE IT
IF YOU WANT ME TO MAKE YOUR OWN SUBJECT PPT CALL ME ON NUMBER
9968821561
Recommended
Welcome
This document discusses the areas of rectangles and triangles. It states that the area of a rectangle is equal to its length multiplied by its breadth. The area of a triangle is equal to one-half of the base multiplied by the height. It then demonstrates that if a line is drawn within a rectangle connecting one vertex to the opposite side, the area of the resulting triangle will be half the area of the original rectangle.
Area of quadrilaterals
This document discusses the area formulas for different types of quadrilaterals. It provides the formulas for calculating the area of rectangles, squares, parallelograms, rhombi, trapezoids, and general quadrilaterals. Examples are given to demonstrate how to use the formulas to calculate areas. The key defines the variables used in the formulas such as base, height, side, and diagonal lengths.
Math Investigation: Pythagorean Theorem
The document illustrates the Pythagorean theorem visually by showing how a larger square can be divided into four triangular regions and a smaller square that together make up the area of the largest square. It provides instructions for readers to cut out and arrange the regions themselves to see how the Pythagorean theorem works. It then suggests two investigation starting points: 1) drawing and cutting out a right triangle with leg lengths of 1 and 2 inches to cover the hypotenuse square, and 2) considering if the same is possible if the smaller leg square is subdivided instead.
017# areas of common shapes
This document discusses how to calculate the areas of common shapes like squares, rectangles, and parallelograms. It states that the area of a square is length times width or length times length, the area of a rectangle is length times width, and the area of a parallelogram is base times perpendicular height or can be calculated using the Pythagorean theorem.
Areas pedro
This document provides formulas for calculating the areas of various shapes including squares, triangles, rectangles, trapezoids, rhombuses, circles, semicircles, cylinders, and semicylinders. It states that the area of a square is side squared, a triangle is half base times height, a rectangle is base times height, a trapezoid is half the sum of the bases times the height, a rhombus can be half the product of the diagonals or base times height, the area of a circle is pi times the radius squared, a semicircle is pi times the radius squared over 2, and the area of a cylinder or semicylinder is pi times the radius squared times the height or half
Mensuration
The document provides formulas for calculating the perimeter and area of basic shapes like squares, rectangles, triangles, parallelograms, and circles. It defines each shape and gives the formula for finding their perimeter and area in terms of common properties like sides, width, length, base, height, radius, and diameter. For example, the perimeter of a square is defined as 4 times the length of one side, and the area of a square is the length of one side squared.
C4 EDEXCEL HELP
The document provides revision notes on various mathematics topics including:
1) Binomial expansions, partial fractions, trigonometry formulas, and techniques for integration like volumes of revolution.
2) Parametric equations, vectors, vector equations, planes, and differential equations.
3) Details are given for solving problems involving these topics, such as using compound angle formulas, rewriting algebraic fractions as partial fractions, and separating variables to solve first-order differential equations.
Mensuration
A MENSURATIO PPT WHICH GIVES EXACT KNOWLEDGE MUST SEE IT
IF YOU WANT ME TO MAKE YOUR OWN SUBJECT PPT CALL ME ON NUMBER
9968821561
Maths project
The document discusses different geometry concepts related to mensuration including cubes, cuboids, cylinders, cones, spheres and hemispheres. It defines key terms like total surface area, lateral surface area and volume. Formulas are provided to calculate surface areas and volumes of cubes, cuboids, cylinders, cones, spheres and hemispheres. Examples are given to illustrate concepts like hollow cylinders and spherical shells.
April 15, 2015
The document provides instruction on completing the square, a method for solving quadratic equations that are not factorable. It explains the five steps: 1) divide by the leading coefficient, 2) rewrite in the form ax + by = c, 3) find half of b and square it, adding it to both sides, 4) rewrite the perfect square trinomial and take the square root of both sides, and 5) solve for x. An example is worked through applying these steps to solve the equation x^2 - 16x + 15 = 0. The document also reviews graphing quadratic functions and finding solutions, axis of symmetry, vertex and y-intercepts.
Alg2 lesson 13-1
This document defines six trigonometric ratios and their abbreviations. It provides examples of using trigonometric functions to solve for unknown sides and angles of right triangles. Specifically, it gives the definitions of sine, cosine, tangent, cosecant, secant, and cotangent. It also reviews trigonometric identities like SOH CAH TOA and provides 4 examples of using trigonometric functions and identities to solve right triangles.
India's narayan-pandit[1]
Narayan Pandit was an Indian mathematician from the 14th century Kerala school of astronomy and mathematics. He wrote two important mathematical texts: Ganit Kaumudi, an arithmetic treatise, and Bijaganita Vatamsa, an algebraic treatise. Ganit Kaumudi covered topics in geometry including formulas for triangles, circles, and areas of triangles, quadrilaterals, and cyclic quadrilaterals. It also described Narayana's original results on the circumradius of circles and presented methods for finding rational triangles and integral triangles with nearly equal sides.
Right triangle problems
This document discusses solving right triangle problems using trigonometric functions like sine, cosine, and tangent. It provides examples of right triangles where certain sides or angles are given and asks the reader to solve for unknown values. Applications discussed include finding the length of a pole or distance across a river using right triangles defined by angles of elevation or depression. Bearings and courses of ships are also defined and used in an example of finding distance and bearings between two moving ships after a period of time.
Area Of Quadrilaterals
This document discusses the formulas to calculate the areas of different quadrilaterals. It provides the area formulas for rectangles as length x width, squares as side x side, parallelograms as base x height, rhombuses as half the product of the diagonals, and trapezoids as half the sum of the bases multiplied by the height.
Formular
The document provides formulas and definitions for various topics in geometry, algebra, trigonometry, and other areas of mathematics. Some of the key information presented includes formulas for calculating the volume and surface area of geometric solids like cubes, cylinders, and cones. In algebra, formulas are given for exponent rules, factoring, and solving equations. Trigonometry formulas define the trigonometric functions and include basic identities. Graphs of common functions like linear, quadratic, and trigonometric functions are also depicted.
Obj. 44 Triangles and Quadrilaterals
The document discusses formulas for finding the areas of triangles and quadrilaterals. It provides the formulas for finding the areas of rectangles, parallelograms, triangles, trapezoids, rhombi, squares, and kites. Examples are given for using the formulas to calculate heights and areas of different shapes.
Math12 lesson5
The document summarizes the key properties and graphs of six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. For each function, it discusses amplitude, period, zeros or asymptotes, and shows an example graph over one period. It also covers transformations of trig functions that change amplitude, period, phase and vertical shifts.
Trigonometry - Strand 3
The document provides information about trigonometry topics for a foundation level class including: solving problems using right triangles, the Pythagorean theorem, defining trig functions like sine, cosine, and tangent for all angle values, graphing periodic trig functions like sine and cosine waves, and noting that the tangent graph has periods of 180 degrees and vertical asymptotes every 90 degrees.
5.13.1 Area of Circles, Sectors, and Quads
Develop and use formulas to find the areas of circles and sectors.
Develop and use formulas to find the areas of special quadrilaterals
Test yourself unit 3 foundation qs
This document provides formulas and definitions for geometry and math concepts covered in Unit 3. It includes formulas for calculating the area of shapes like triangles, parallelograms, trapezoids, and circles. It also includes formulas for volume of cubes, cuboids, prisms, cylinders, as well as definitions for terms like congruent, similar, perpendicular, and more. The document is a comprehensive study guide listing key formulas and explanations for foundational geometry and math topics.
Powerpoint(pythagorean theorem)
The document discusses the Pythagorean theorem and its applications to right triangles. It defines the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. It provides examples of using the Pythagorean theorem to calculate missing sides of right triangles when given two sides. Exercises at the end ask the reader to apply the Pythagorean theorem to find missing sides of right triangles.
11.2 Area of Triangles and Quads
The document discusses formulas for finding the areas of various shapes. It provides formulas for finding the areas of triangles, parallelograms, trapezoids, rhombi, kites, rectangles, and equilateral triangles. An example is worked through to find the area of a parallelogram and the length of one side of a kite given the length of the other side and the total area.
Class2 geometry and trigonometry
This document provides information about trigonometry and geometry concepts. It discusses Pythagorean triples and the Pythagorean theorem. It then explains trigonometric functions like sine, cosine, and tangent. It also discusses generating Pythagorean triples and using trig functions for angles greater than 90 degrees. Finally, it covers topics like the unit circle, solving trigonometric equations, triangulation using the sine and cosine rules, and properties of circles, triangles, polygons, and other geometric shapes.
Obj. 29 Area Formulas
The document discusses formulas for calculating the areas of various shapes including triangles, quadrilaterals, parallelograms, trapezoids, rhombi, squares, and kites. It provides the area formulas, explains how to derive the formulas for parallelograms, triangles, and trapezoids from rectangles, and includes examples of using the formulas to calculate heights and areas.
MathsQuadrilateral.pdf
- A quadrilateral is a four-sided polygon. There are several types of quadrilaterals defined by their properties: square, rectangle, parallelogram, rhombus, trapezoid, kite, and irregular quadrilateral.
- The sum of the interior angles of any quadrilateral is always 360 degrees.
- Formulas are provided to calculate the perimeter and area of each quadrilateral type based on given side lengths or angles. This includes using side lengths, base and height, diagonals, and trigonometric functions like sine.
- An irregular quadrilateral can be divided into two triangles to calculate its total area using the individual triangle area formulas.
MATHS IN-CLASS ACTIVITY SLIDES
- A quadrilateral is a four-sided polygon. There are several types of quadrilaterals defined by their properties: square, rectangle, parallelogram, rhombus, trapezoid, kite, and irregular quadrilateral.
- The sum of the interior angles of any quadrilateral is always 360 degrees.
- Formulas are provided to calculate the perimeter and area of different quadrilaterals depending on given side lengths or angles. This includes formulas for square, rectangle, parallelogram, rhombus, trapezoid, kite, and methods described to find the area of an irregular quadrilateral by dividing it into triangles.
- A cyclic quadrilateral has the special property
Regression planar
Planar regression calculates the best fit plane through 3 or more data points by minimizing the residuals between the plane and points using least squares minimization. The equation for the plane is determined by taking the derivative of the residuals equation and setting it equal to zero, yielding a system of equations that can be expressed in matrix form and solved for the plane coefficients.
Applications of integration
This document discusses methods for calculating volumes using integration. There are three main methods - disc, washer, and shell. The disc and washer methods use the formula πr2h to find the volume of revolution around an axis, with disc being for hollow shapes and washer for shapes within other shapes. The shell method uses 2πrh and is primarily used for revolutions around the y-axis. It is important to determine the correct graph and limits of integration based on the problem.
Flash cards AoI
This document contains flash cards to help learn concepts related to integrals and approximation methods. The flash cards cover whether different approximation methods overvalue or undervalue a function, which approximation method provides the best estimate, using the disc method to find the area under a curve between bounds, and using the washer method to find the area between two curves where one is always on top between given bounds.
Application of integrals flashcards
This document provides an overview of different integration methods for calculating volumes rotated around axes:
1) The Washer Method calculates volumes between two curves rotated around an axis by integrating the area of circular cross-sections.
2) The Inverse Area Method finds areas between curves by rewriting the functions in inverse form and integrating the top curve minus the bottom curve.
3) The Shell Method, similar to Washer, switches the variable of integration depending on the axis of rotation and uses the thickness and height of shell-shaped cross-sections.
4) Other methods discussed include Left Rectangular Approximation and the Disk Method for calculating the volume of a single rotated function using circular cross-sections.
More Related Content
What's hot
Maths project
The document discusses different geometry concepts related to mensuration including cubes, cuboids, cylinders, cones, spheres and hemispheres. It defines key terms like total surface area, lateral surface area and volume. Formulas are provided to calculate surface areas and volumes of cubes, cuboids, cylinders, cones, spheres and hemispheres. Examples are given to illustrate concepts like hollow cylinders and spherical shells.
April 15, 2015
The document provides instruction on completing the square, a method for solving quadratic equations that are not factorable. It explains the five steps: 1) divide by the leading coefficient, 2) rewrite in the form ax + by = c, 3) find half of b and square it, adding it to both sides, 4) rewrite the perfect square trinomial and take the square root of both sides, and 5) solve for x. An example is worked through applying these steps to solve the equation x^2 - 16x + 15 = 0. The document also reviews graphing quadratic functions and finding solutions, axis of symmetry, vertex and y-intercepts.
Alg2 lesson 13-1
This document defines six trigonometric ratios and their abbreviations. It provides examples of using trigonometric functions to solve for unknown sides and angles of right triangles. Specifically, it gives the definitions of sine, cosine, tangent, cosecant, secant, and cotangent. It also reviews trigonometric identities like SOH CAH TOA and provides 4 examples of using trigonometric functions and identities to solve right triangles.
India's narayan-pandit[1]
Narayan Pandit was an Indian mathematician from the 14th century Kerala school of astronomy and mathematics. He wrote two important mathematical texts: Ganit Kaumudi, an arithmetic treatise, and Bijaganita Vatamsa, an algebraic treatise. Ganit Kaumudi covered topics in geometry including formulas for triangles, circles, and areas of triangles, quadrilaterals, and cyclic quadrilaterals. It also described Narayana's original results on the circumradius of circles and presented methods for finding rational triangles and integral triangles with nearly equal sides.
Right triangle problems
This document discusses solving right triangle problems using trigonometric functions like sine, cosine, and tangent. It provides examples of right triangles where certain sides or angles are given and asks the reader to solve for unknown values. Applications discussed include finding the length of a pole or distance across a river using right triangles defined by angles of elevation or depression. Bearings and courses of ships are also defined and used in an example of finding distance and bearings between two moving ships after a period of time.
Area Of Quadrilaterals
This document discusses the formulas to calculate the areas of different quadrilaterals. It provides the area formulas for rectangles as length x width, squares as side x side, parallelograms as base x height, rhombuses as half the product of the diagonals, and trapezoids as half the sum of the bases multiplied by the height.
Formular
The document provides formulas and definitions for various topics in geometry, algebra, trigonometry, and other areas of mathematics. Some of the key information presented includes formulas for calculating the volume and surface area of geometric solids like cubes, cylinders, and cones. In algebra, formulas are given for exponent rules, factoring, and solving equations. Trigonometry formulas define the trigonometric functions and include basic identities. Graphs of common functions like linear, quadratic, and trigonometric functions are also depicted.
Obj. 44 Triangles and Quadrilaterals
The document discusses formulas for finding the areas of triangles and quadrilaterals. It provides the formulas for finding the areas of rectangles, parallelograms, triangles, trapezoids, rhombi, squares, and kites. Examples are given for using the formulas to calculate heights and areas of different shapes.
Math12 lesson5
The document summarizes the key properties and graphs of six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. For each function, it discusses amplitude, period, zeros or asymptotes, and shows an example graph over one period. It also covers transformations of trig functions that change amplitude, period, phase and vertical shifts.
Trigonometry - Strand 3
The document provides information about trigonometry topics for a foundation level class including: solving problems using right triangles, the Pythagorean theorem, defining trig functions like sine, cosine, and tangent for all angle values, graphing periodic trig functions like sine and cosine waves, and noting that the tangent graph has periods of 180 degrees and vertical asymptotes every 90 degrees.
5.13.1 Area of Circles, Sectors, and Quads
Develop and use formulas to find the areas of circles and sectors.
Develop and use formulas to find the areas of special quadrilaterals
Test yourself unit 3 foundation qs
This document provides formulas and definitions for geometry and math concepts covered in Unit 3. It includes formulas for calculating the area of shapes like triangles, parallelograms, trapezoids, and circles. It also includes formulas for volume of cubes, cuboids, prisms, cylinders, as well as definitions for terms like congruent, similar, perpendicular, and more. The document is a comprehensive study guide listing key formulas and explanations for foundational geometry and math topics.
Powerpoint(pythagorean theorem)
The document discusses the Pythagorean theorem and its applications to right triangles. It defines the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. It provides examples of using the Pythagorean theorem to calculate missing sides of right triangles when given two sides. Exercises at the end ask the reader to apply the Pythagorean theorem to find missing sides of right triangles.
11.2 Area of Triangles and Quads
The document discusses formulas for finding the areas of various shapes. It provides formulas for finding the areas of triangles, parallelograms, trapezoids, rhombi, kites, rectangles, and equilateral triangles. An example is worked through to find the area of a parallelogram and the length of one side of a kite given the length of the other side and the total area.
Class2 geometry and trigonometry
This document provides information about trigonometry and geometry concepts. It discusses Pythagorean triples and the Pythagorean theorem. It then explains trigonometric functions like sine, cosine, and tangent. It also discusses generating Pythagorean triples and using trig functions for angles greater than 90 degrees. Finally, it covers topics like the unit circle, solving trigonometric equations, triangulation using the sine and cosine rules, and properties of circles, triangles, polygons, and other geometric shapes.
Obj. 29 Area Formulas
The document discusses formulas for calculating the areas of various shapes including triangles, quadrilaterals, parallelograms, trapezoids, rhombi, squares, and kites. It provides the area formulas, explains how to derive the formulas for parallelograms, triangles, and trapezoids from rectangles, and includes examples of using the formulas to calculate heights and areas.
MathsQuadrilateral.pdf
- A quadrilateral is a four-sided polygon. There are several types of quadrilaterals defined by their properties: square, rectangle, parallelogram, rhombus, trapezoid, kite, and irregular quadrilateral.
- The sum of the interior angles of any quadrilateral is always 360 degrees.
- Formulas are provided to calculate the perimeter and area of each quadrilateral type based on given side lengths or angles. This includes using side lengths, base and height, diagonals, and trigonometric functions like sine.
- An irregular quadrilateral can be divided into two triangles to calculate its total area using the individual triangle area formulas.
MATHS IN-CLASS ACTIVITY SLIDES
- A quadrilateral is a four-sided polygon. There are several types of quadrilaterals defined by their properties: square, rectangle, parallelogram, rhombus, trapezoid, kite, and irregular quadrilateral.
- The sum of the interior angles of any quadrilateral is always 360 degrees.
- Formulas are provided to calculate the perimeter and area of different quadrilaterals depending on given side lengths or angles. This includes formulas for square, rectangle, parallelogram, rhombus, trapezoid, kite, and methods described to find the area of an irregular quadrilateral by dividing it into triangles.
- A cyclic quadrilateral has the special property
Regression planar
Planar regression calculates the best fit plane through 3 or more data points by minimizing the residuals between the plane and points using least squares minimization. The equation for the plane is determined by taking the derivative of the residuals equation and setting it equal to zero, yielding a system of equations that can be expressed in matrix form and solved for the plane coefficients.
What's hot (19)
Viewers also liked
Applications of integration
This document discusses methods for calculating volumes using integration. There are three main methods - disc, washer, and shell. The disc and washer methods use the formula πr2h to find the volume of revolution around an axis, with disc being for hollow shapes and washer for shapes within other shapes. The shell method uses 2πrh and is primarily used for revolutions around the y-axis. It is important to determine the correct graph and limits of integration based on the problem.
Flash cards AoI
This document contains flash cards to help learn concepts related to integrals and approximation methods. The flash cards cover whether different approximation methods overvalue or undervalue a function, which approximation method provides the best estimate, using the disc method to find the area under a curve between bounds, and using the washer method to find the area between two curves where one is always on top between given bounds.
Application of integrals flashcards
This document provides an overview of different integration methods for calculating volumes rotated around axes:
1) The Washer Method calculates volumes between two curves rotated around an axis by integrating the area of circular cross-sections.
2) The Inverse Area Method finds areas between curves by rewriting the functions in inverse form and integrating the top curve minus the bottom curve.
3) The Shell Method, similar to Washer, switches the variable of integration depending on the axis of rotation and uses the thickness and height of shell-shaped cross-sections.
4) Other methods discussed include Left Rectangular Approximation and the Disk Method for calculating the volume of a single rotated function using circular cross-sections.
Calculus Application of Integration
The document discusses various methods for calculating the volume of solids of revolution using integration, including the disk method, washer method, and cylindrical shell method. It provides examples of applying each method to find the volume of solids formed by revolving regions between curves about axes. The disk method approximates the solid as disks of thickness dx or dy. The washer method accounts for solids with holes by using washers of thickness dx or dy and inner and outer radii. Examples demonstrate setting up and evaluating the integrals to find the volumes.
Integration application (Aplikasi Integral)
Materi kuliah tentang Aplikasi Integral. Cari lebih banyak mata kuliah Semester 1 di: http://muhammadhabibielecture.blogspot.com/2014/12/kuliah-semester-1-thp-ftp-ub.html
Applications of Definite Intergral
This document provides an introduction to using definite integrals to calculate volumes, lengths of curves, centers of mass, surface areas, work, and fluid forces. It discusses calculating volumes through slicing solids and rotating areas about an axis. Examples are provided for finding the volumes of pyramids, wedges, and solids of revolution. It also discusses using integrals to find curve lengths, circle circumferences, and moments and centers of mass for various objects. Surface areas of revolution and fluid pressures are also explained.
Final Project Powerpoint
This document discusses different methods for approximating the area under a curve using rectangular approximations. It explains left, right, and middle rectangular approximations and which is most accurate. Middle rectangular approximation is most accurate as it takes the average of the left and right approximations. The document also discusses how rectangular approximation is linked to finding the actual area under a curve and provides shortcuts for calculating different rectangular approximations using a calculator. Finally, it discusses various area and volume methods like disks, washers, shells, and integrals.
ppt on application of integrals
this is the ppt on application of integrals, which includes-area between the two curves , volume by slicing , disk method , washer method, and volume by cylindrical shells,.
this is made by dhrumil patel and harshid panchal.
Viewers also liked (8)
Similar to Applications of integration
Maths project on mensuration class 8 by indira singh
This document is a math project on mensuration submitted by Indira Singh of class VIIIA. It defines mensuration as the branch of mathematics dealing with measuring the areas and volumes of geometric figures. It then provides definitions of area and perimeter. The bulk of the document discusses different geometric shapes like squares, rectangles, triangles, parallelograms, trapezoids, rhombuses, cubes, cuboids and cylinders. It provides the number of sides, vertices, formulas to calculate the area and perimeter of two-dimensional shapes. It also gives the formulas to calculate the surface area and volume of three-dimensional solids.
Ap calculus warm up 8.27.13
To find the limit of a function as x approaches positive or negative infinity, examine the degrees of the numerator and denominator polynomials. If the degrees are equal, the limit is the ratio of the leading coefficients; if the numerator degree is greater, the limit does not exist; if the denominator degree is greater, the limit is 0. Rational functions can have horizontal or vertical asymptotes. Examples show evaluating limits of rational functions by comparing degrees and leading coefficients as x approaches infinity. Practice problems apply these techniques to additional rational function limits.
Prestation_ClydeShen
This document discusses graph colouring and graph dynamical systems. It begins with an overview of graph theory concepts like graphs, graph colouring, and the graph colouring problem. It then discusses deterministic finite automata and introduces graph-cellular automata as a type of graph dynamical system. Specific examples of graph-cellular automata on linear graphs, circle graphs, tree graphs, wheel graphs, and Peterson graphs are analyzed. The results show that linear graphs, circle graphs, and tree graphs reach a stable coloring, while wheel graphs and Peterson graphs result in a loop.
Karnaugh maps
This document discusses Karnaugh maps, which are a graphical technique for simplifying Boolean expressions into minimal sum-of-products form. It covers how to:
1) Construct Karnaugh maps for 2, 3, and 4 variable expressions from truth tables or sums of minterms.
2) Group adjacent squares containing common literals to find the simplest product terms.
3) Read the minimal sum-of-products form from the groupings on the map.
Examples are provided to demonstrate how to simplify expressions and construct Karnaugh maps.
Matlab tutorial 2
MATLAB Tutorial 2 provides an overview of built-in MATLAB functions across several categories: math functions, complex numbers, matrices algebra, plotting, and data analysis. It includes examples of using functions to find absolute values, square roots, logarithms, factors, and other operations. It also demonstrates how to perform matrix operations like inverses, determinants, and special matrices. The document shows how to create basic and advanced plots, including polar, 3D, and subplot graphs.
Week6.ppt
The document discusses various 2D geometric transformations including reflections, shears, and their matrix representations. It also covers 2D viewing pipelines including coordinate systems, clipping techniques like Cohen-Sutherland and Liang-Barsky line clipping, and Sutherland-Hodgeman polygon clipping. Reflections are described as 180 degree rotations about an axis. Shearing shifts points along an axis proportional to their coordinate on the other axis. Clipping algorithms discard or shorten line/polygon segments that fall outside the clip region.
Lecture co4 math21-1
This document provides instruction on applying derivatives to solve various types of application problems. It begins by outlining objectives of analyzing and solving application problems involving derivatives as instantaneous rates of change or tangent line slopes. Examples of application problems covered include writing equations of tangent and normal lines, curve tracing, optimization problems, and related rates problems involving time rates. The document then provides definitions and examples of using derivatives to find slopes of curves and tangent lines. It also covers concepts like concavity, points of inflection, maxima/minima, and solving optimization problems using derivatives. Finally, it gives examples of solving related rates problems involving time-dependent variables.
Linear functions
The document discusses different forms and methods for graphing linear functions, including:
1) Standard, point-slope, and slope-intercept forms of linear equations.
2) Converting between these forms by solving for slope and y-intercept.
3) Graphing linear functions by finding the slope from two points using the point-slope form, making a table of x-y values, or finding the x- and y-intercepts.
4) Determining if two functions are parallel or perpendicular based on having equal or reciprocal slopes.
Linear functions
The document discusses different forms and methods for graphing linear functions, including:
1) The standard, point-slope, and slope-intercept forms of linear equations.
2) Converting between these forms by solving for slope and y-intercept.
3) Graphing linear functions by finding the slope from two points using the point-slope form, making a table of x-y values, or finding the x- and y-intercepts.
4) Determining if two functions are parallel or perpendicular based on having equal or reciprocal slopes.
GEOMETRY.DOCX
Polygons can be regular or irregular. Regular polygons have all sides and angles equal, while irregular polygons do not. Common polygons include triangles, quadrilaterals, pentagons, and hexagons. Polygons are 2D shapes with straight sides. 3D shapes include polyhedra with flat faces that are polygons, as well as non-polyhedral shapes like cylinders, cones, and spheres. Polyhedra include the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) as well as prisms and pyramids. Prisms have two identical polygon bases and parallelogram sides, while pyramids have one polygon base and triangular sides
Inverse trig functions
The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.
Graphing quadratic equations
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
Similar to Applications of integration (12)
Maths project on mensuration class 8 by indira singh
Maths project on mensuration class 8 by indira singh
Recently uploaded
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...National Information Standards Organization (NISO)
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.DRUGS AND ITS classification slide share
Any substance (other than food) that is used to prevent, diagnose, treat, or relieve symptoms of a
disease or abnormal condition
The Diamonds of 2023-2024 in the IGRA collection
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Digital Artifact 1 - 10VCD Environments Unit
Digital Artifact 1 - 10VCD Environments Unit - NGV Pavilion Concept Design
Main Java[All of the Base Concepts}.docx
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Community pharmacy- Social and preventive pharmacy UNIT 5
Covered community pharmacy topic of the subject Social and preventive pharmacy for Diploma and Bachelor of pharmacy
Hindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Advanced Java[Extra Concepts, Not Difficult].docx
This is part 2 of my Java Learning Journey. This contains Hashing, ArrayList, LinkedList, Date and Time Classes, Calendar Class and more.
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityAkanksha trivedi rama nursing college kanpur.
Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcবাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Setup Warehouse & Location in Odoo 17 Inventory
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Aberdeen
How to Make a Field Mandatory in Odoo 17
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Recently uploaded (20)
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Community pharmacy- Social and preventive pharmacy UNIT 5
Community pharmacy- Social and preventive pharmacy UNIT 5
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
How to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 Inventory
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Liberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdf
Applications of integration
- 1. Group 4 (Alex Gagliardi) Applications of Integration
- 2. Area Under a Curve: Rectangular Approximation
- 3. Left Rectangular Approx. Method (LRAM) Right Rectangular Approx. Method (RRAM) Midpoint Rectangular Approx. Method (MRAM n= number of rectangles Delta x= width of rectangle in the interval [a,b] MRAM is the most accurate because the area that you are finding is more accurate compared to RRAM and LRAM.
- 4. Area Methods
- 5. X-axis Original equations: y= Left Right In terms of … Two curves: top bottom One curve below X-axis: -
- 6. Y-axis Original Equations Y- axis Bottom Top In terms of y: … Outer – inner /or right – left One curve on left y-axis:
- 7. Inverse Different graph y=x; x y; solve for y X-axis: left right … terms of x Top – bottom *HINT* x=!!!!
- 9. Disk Method Revolved around the x-axis Disk- “cylinders” r= the function value *always square the radius a.k.a the function value
- 10. Washer Method Y-axis in terms of y: Outer curve – inner curve dy or dx
- 11. Shell Method Rotate about the y-axis top curve -bottom curve In terms of x dy or/ dx
- 12. Area is connected to Rectangular approximation method because both involve finding the area under a curve or within a curve. How is Area linked to Rectangular Approximation Method?