Solving volumes using cross sectional areas
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This document provides an explanation of how to calculate the volume of a solid with a cross-sectional area that changes along one axis. It gives the example of finding the volume of a solid where the cross-section is a triangle perpendicular to the x-axis, with base that varies as a function of x from 0 to 4. The document provides the formula for the area of the triangle as a function of x, and the integral required to calculate the volume by summing the areas of each cross-sectional slice from x=0 to x=4.
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![Solving Volume of a Cross-Sectional
Solid
•Base: y 1 2
x 2 x
y 2
1
y x2 2 x •Cross Sectional Shapes:
2
•Triangles perpendicular to the x-
axis.
•Interval: [0,4]
x
•Triangle Base: Triangle Height = 2:3
•Triangle Base= (2*(Triangle Height))/
3
1 3
•Area of the Whole Triangle: ( Base * 2)( )
2 4
•Function for Solving the Volume of the
Solid: 4
3 1 2
( x 2 x ) 2 dx
0
3 2](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Rene Descartes invented the Cartesian coordinate system which relates points on a plane to pairs of real numbers. The system maps points onto a grid defined by perpendicular x and y axes, dividing the plane into four quadrants. Ordered pairs of real numbers (x, y) uniquely identify points by describing their position relative to the intersecting axes. For example, the point P is located at coordinates (2, 3) and point Q at (-5, -4).
Ellipses
The document contains information about ellipses including:
- An ellipse is defined as the set of all points where the sum of the distances from two fixed points (foci) is a constant.
- The major and minor axes, vertices, foci, and center of an ellipse are identified.
- The standard equation for an ellipse is provided and examples of graphing ellipses from equations are shown.
- One application of ellipses mentioned is the whispering gallery, where sound travels between the two foci of an elliptical ceiling.
Ap calc intro (slideshare version) 2014
This document provides an overview of the AP Calculus AB course taught by Mr. Eick. The course will cover calculus concepts like functions, derivatives, integrals, and differential equations over two semesters. Students are expected to complete summer work, use a graphing calculator, and study independently. The teacher emphasizes working hard, asking questions, and obtaining a passing score on the AP exam.
Pre calculus warm up 4.21.14
This document defines and provides examples of permutations and combinations. Permutations refer to arrangements of objects where order matters, while combinations refer to collections of objects where order does not matter. The key formulas discussed are permutations of n objects taken r at a time (nPr) and combinations of n objects taken r at a time (nCr). Several word problems demonstrate calculating permutations and combinations in different contexts such as arranging letters, selecting teams or committees, and more.
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Elementos finitos
The document provides instructions for solving an ordinary differential equation using the finite element method. Specifically, it asks to:
1. Discretize the given differential equation over 7 equally spaced subintervals between 0 and 4.
2. Assemble the matrix equation from the discretized problem and calculate the load vector.
3. Solve the matrix equation to find the approximated solution at each node.
It then provides the details for constructing the stiffness matrix based on the given differential equation and basis functions over each subinterval.
Day 2 graphing linear equations
This document provides instructions for graphing linear equations. It begins with examples of solving linear equations algebraically. Students are then introduced to key properties of linear equations: they contain two variables and graph as straight lines. The document demonstrates graphing various linear equations by plotting their solution sets as points and connecting them with a straight line. It concludes by asking students to reflect on similarities and differences between the graphed linear equations.
BBMP1103 - Sept 2011 exam workshop - part 4
This document is a summary of Part 4 of a mathematics exam preparation workshop on integration. It provides examples and step-by-step solutions for two exam questions involving integration. The first question from May 2010 involves integrating expressions including (3x3 - 3x2 + x). The second question from January 2010 involves integrating (a) 1/x3, (b) x/x3 - 2, and (c) (4x3 - 2x). The solutions show the integration steps and resulting expressions for each part.
Pc12 sol c04_4-4
This document provides examples and exercises on determining composite functions from given functions. It includes:
- Examples of determining possible functions f and g for composite functions like y = (x + 4)2 and y = √x + 5.
- A table sketching and finding domains and ranges for various composite functions like y = f(f(x)) and y = f(g(x)) given functions f(x) and g(x).
- Exercises to determine composite functions f(g(x)) and possible functions f, g, and h for more complex functions like y = x2 - 6x + 5.
- Questions about restrictions on variables and domains for composite functions
Pc12 sol c04_4-1
1. The document provides graphs and equations for functions f(x), g(x), and their combinations. It asks the reader to sketch graphs, determine domains and ranges, and solve related problems.
2. The key combinations are addition, subtraction, multiplication, and division of f(x) and g(x). Their domains and ranges are identified from the original function graphs.
3. For a combination like f(x) + g(x), the domain is the same as the more restrictive of the two original functions, while the range includes all outputs equal to or greater than the original function ranges.
ฟังก์ชัน(function)
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
Day 3 graphing linear equations
This document provides instructions for graphing linear equations. It begins with a "Do Now" activity to find and correct mistakes in a sample function table and graph. It then shows examples of graphing different linear equations by plotting points from their function tables on coordinate grids. The document concludes with a prompt to explain to an absent student how to determine from a graph if a mistake was made in a problem.
Pc12 sol c04_cp
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
Pc12 sol c04_cp
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
Day 1 intro to functions
The document discusses using functions to find output values from input values. It provides examples of functions in the form of y=fx(x) and has students complete function tables and plot points for various functions. It discusses how the Rube Goldberg cartoon from the beginning uses an input, output, and rule to demonstrate a function.
Chapter 07
This document contains 14 math problems involving calculating the area under curves using definite integrals. The problems include finding the area under functions such as x2, √x, sec2x, and siny from given bounds. The areas are calculated and expressed as fractions or simplified numeric values.
Pc12 sol c04_review
This document provides solutions to review problems involving combining functions through addition, subtraction, multiplication, division, and composition. Some key examples include:
- Sketching the graphs of f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x) given the graphs of f(x) and g(x)
- Writing explicit equations for combinations of functions and determining their domains and ranges
- Evaluating composite functions like f(g(x)) and g(f(x)) given definitions of f(x) and g(x)
- Determining if two functions are inverses using their compositions
[4] num integration
The document discusses numerical integration methods for calculating ship geometrical properties. It introduces trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule for numerical integration. Simpson's 1st rule is recommended for calculating properties like waterplane area, sectional area, submerged volume, and centers of floatation and buoyancy which involve integrating curves related to the ship's shape. Detailed steps are provided for applying Simpson's 1st rule to calculate these properties numerically.
Answer to selected_miscellaneous_exercises
This document contains solutions to selected miscellaneous exercises involving calculus concepts like derivatives, logarithms, and integrals. Some key points:
- Exercise 8 involves taking the derivative of an expression involving logarithms and solving for n. The answer is n = 20.
- Exercise 9 proves an identity involving natural logarithms using derivatives.
- Exercise 14 finds the derivative of an expression involving a logarithm.
Chapter 04
This document contains 26 math problems involving derivatives of logarithmic, exponential, and inverse trigonometric functions. The problems include finding derivatives of expressions, setting up and solving differential equations, and determining relationships between derivatives.
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The document defines and provides examples of several types of functions including:
1) Constant functions where f(x) = a for all values of x.
2) Linear functions of the form f(x) = ax + b.
3) Quadratic functions of the form f(x) = ax2 + bx + c.
4) Polynomial functions which are the sum of terms with variables raised to various powers.
Inequalities quadratic, fractional & irrational form
To solve quadratic, fractional, irrational, and absolute value inequalities, one should:
1. Make the right-hand side zero by shifting terms to the left-hand side
2. Fully factorize the left-hand side to find critical values
3. Draw a sign diagram for the left-hand side using the critical values
4. Determine the range of values for the variable based on the sign diagram.
ฟังก์ชัน(function)
The document defines and provides examples of different types of functions:
1. Constant functions where f(x) = a for all values of x (e.g. f(x) = 2).
2. Linear functions of the form f(x) = ax + b (e.g. f(x) = 5x+3).
3. Quadratic functions of the form f(x) = ax2 + bx + c (e.g. f(x) = 3x2 + 2x + 1).
5.9 notes
This document discusses how to evaluate powers of products and quotients. It states that to evaluate a product to a power, the power is distributed to each factor. To evaluate a quotient to a power, the power is distributed to both the numerator and denominator. Several examples are provided of simplifying expressions involving products and quotients raised to powers. The document concludes with a list of additional practice problems for students.
Lesson 54
The document provides guidelines for graphing polynomial and rational functions. It discusses the key features of graphs of quadratic, cubic, quartic and quintic polynomials. It then discusses how to graph rational functions by identifying intercepts, asymptotes, discontinuities and using sign analysis to determine the positive and negative portions of the graph. An example rational function is graphed as an illustration.
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Inequalities quadratic, fractional & irrational form
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Additional review problems
The number of jobs grows exponentially over time based on the existing number of jobs. Given there were 15 million jobs initially and 15.1 million jobs 3 months later, the model estimates there will be 15.8 million jobs in two years.
A population of critters increases directly proportional to the difference between its maximum size and current size. Given the population was initially 1800 and 1750 after 5 years, the model estimates the population will be 1918 critters after 20 years.
Savanah presentatiom
This document defines and describes several geometric shapes including linear pairs of angles, octagons, parallel lines, parallelograms, arcs, circles, trapezoids, and isosceles triangles. It provides brief definitions for each shape stating their key properties such as number of sides, angles, and relationships between sides or angles.
Properties of logarithms
The document discusses various properties of logarithms including the product, quotient, and power properties. It also covers expanding logarithmic expressions, changing bases using the change of base formula, and provides an example using logarithms to calculate an increase in decibels when sound intensity doubles. Worked examples are provided to demonstrate evaluating logarithmic expressions and using properties to simplify or expand logarithmic terms.
Finding volume of a solid using cross sectional areas
This document provides instructions to calculate the volume of a solid with a given base and cross sections of equilateral triangles. The base is defined by the equation y=16-x^2 from x=4 to x=4. To find the volume, the document calculates the area of each cross section triangle using the length of its sides, and integrates the area function from x=4 to x=4. The final volume is given as 64π/3.
Approximating area
This document discusses several methods for approximating the area under a curve, including using rectangles with right endpoints, left endpoints, and midpoints. It provides formulas for the right endpoint, left endpoint, and midpoint approximations. It also presents an example problem demonstrating how to use these methods to compute the area under a curve, and provides practice problems for the reader to try applying the techniques.
Tessellation project
This document provides an introduction to a student project on tessellations. It discusses tessellations created by Dutch artist M.C. Escher and the different types of tessellations that can be made through translations, glide reflections, and rotations of shapes. The document provides examples of each type and suggestions for students to design their own tessellation template and finished tessellation project. Students are expected to turn in their completed tessellation and template for grading by a due date.
Approximating area
This document discusses several methods for approximating the area under a curve in calculus, including using rectangles with the right endpoint, left endpoint, and midpoint formulas to divide the area into sections and add them together. It also presents the power sums formula and works through an example problem of using the right endpoint rectangle approximation with N sections to find the area under a given curve and taking the limit as N approaches infinity.
Properties of rational exponents
This document discusses properties of rational exponents and radicals. It outlines 6 properties of rational exponents including how to simplify expressions with rational exponents. It also outlines 2 properties of radicals including how the radical of a variable changes when raised to an even or odd power. The document provides examples of applying these properties and a rule for when absolute value bars are not needed for radicals containing variables.
Evaluate nth roots and use rational exponents
This document discusses algebra concepts including:
- Exponents and their properties such as 32 = 9 and 102 = 100.
- Roots and their properties depending on if the root is even or odd and if the radicand is positive, negative or zero.
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Ap calculus sem exam item analysis
The document summarizes the results of a semester 1 exam with 35 multiple choice questions. It provides the question number, correct response, and percent of students who answered correctly for each question. The summary at the bottom indicates the mean score was 90.5% with a standard deviation of 2.13. The questions most commonly missed were numbers 1, 10, 18, 19, 25, 27, and 35.
3.2 Derivative as a Function
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
Precalc review for calc p pt
This document is a powerpoint presentation that serves as a pre-calculus review quiz covering topics like:
- Definitions of even and odd functions
- Identifying graphs of simple functions like f(x)=x^2
- Logarithmic and trigonometric identities and properties
- Trigonometric function values for common angles
- Algebraic identities for expanding expressions like (a+b)^3
The presentation is interactive, asking questions and providing feedback on slides to test the user's knowledge of important pre-calculus concepts needed for success in calculus. It concludes by stating if the user knows this material, they are prepared to begin studying calculus.
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Finding volume of a solid using cross sectional areas
Finding volume of a solid using cross sectional areas
Solving volumes using cross sectional areas
- 1. SOLVING VOLUMES USING CROSS SECTIONAL AREAS AP Calculus AB/ Christy Sohn (11)
- 2. Solving Volume of a Cross-Sectional Solid •Base: y 1 2 x 2 x y 2 1 y x2 2 x •Cross Sectional Shapes: 2 •Triangles perpendicular to the x- axis. •Interval: [0,4] x •Triangle Base: Triangle Height = 2:3 •Triangle Base= (2*(Triangle Height))/ 3 1 3 •Area of the Whole Triangle: ( Base * 2)( ) 2 4 •Function for Solving the Volume of the Solid: 4 3 1 2 ( x 2 x ) 2 dx 0 3 2
- 3. Solutions 4 Volume ( Area) dx 0 y 4 3 1 2 1 ( x 2 x ) 2 dx y x2 2 x 3 2 2 0 4 5 3 1 4 [( ( x 2 x 2 4 x)dx] 3 0 4 x 7 4 3 1 5 4 ( x x 2 2x2 ) 3 20 7 0 7 7 3 1 4 1 4 [( (4) 5 (4) 2 2(4) ) ( (0) 5 2 (0) 2 2(0) 2 )] 3 20 7 20 7 3 256 512 3 5472 1824 3 ( 32 ) ( ) 3 5 7 3 35 35 90 .2646 90 .26