1523 double integrals
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![Volume and Double Integral
z=f(x,y) ≥ 0
on rectangle
R=[a,b]×[c,d]
S={(x,y,z) in R3
| 0 ≤ z ≤ f(x,y), (x,y) in R}
Volume of S = ?](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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This document provides information on several multivariable calculus topics:
1) Finding maxima and minima of functions of two variables using partial derivatives and the second derivative test.
2) Finding the tangent plane and normal line to a surface.
3) Taylor series expansions for functions of two variables.
4) Standard expansions for common functions like e^x, cosh(x), and tanh(x) using Maclaurin series.
5) Linearizing functions around a point using the tangent plane approximation.
6) Lagrange's method of undetermined multipliers for finding extrema with constraints.
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1) The geometrical meaning of partial derivatives as the slope of the tangent line to a surface.
2) Finding the equation of the tangent plane and normal line to a surface.
3) Taylor's theorem and Maclaurin's theorem for functions with two variables, which can be used to approximate functions and calculate errors.
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2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
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Smart solution un matematika sma 2013 (skl 5.4 aplikasi integral (luas daerah...
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FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docx
FINAL PROJECT, MATH 251, FALL 2015
[The project is Due Monday after the thanks giving recess]
.NAME(PRINT).________________ SHOW ALL WORK. Explain and
SKETCH (everywhere anytime and especially as you try to comprehend the prob-
lems below) whenever possible and/or necessary. Please carefully recheck your
answers. Leave reasonable space between lines on your solution sheets. Number
them and print your name.
Please sign the following. I hereby affirm that all the work in this project was
done by myself ______________________.
1) i) Explain how to derive the representation of the Cartesian coordinates x,y,z
in terms of the spherical coordinates ρ, θ, φ to obtain
(0.1) r =< x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) > .
What are the conventional ranges of ρ, θ, φ?
ii) Conversely, explain how to express ρ, sin(θ), cos(θ), cos(φ), sin(φ) as
functions of x,y,z.
iii) Consider the spherical coordinates ρ,θ, φ. Sketch and describe in your own
words the set of all points x,y,z in x,y,z space such that:
a) 0 ≤ ρ ≤ 1, 0 ≤ θ < 2π, 0 ≤ φ ≤ π b) ρ = 1, 0 ≤ θ < 2π, 0 ≤ φ ≤ π,
c) 0 ≤ ρ < ∞, 0 ≤ θ < 2π, φ = π
4
, d) ρ = 1, 0 ≤ θ < 2π, φ = π
4
,
e) ρ = 1, θ = π
4
, 0 ≤ φ ≤ π. f) 1 ≤ ρ ≤ 2, 0 ≤ θ < 2π, π
6
≤ φ ≤ π
3
.
iv) In a different set of Cartesian Coordinates ρ, θ, φ sketch and describe in your
own words the set of points (ρ, θ, φ) given above in each item a) to f). For example
the set in a) in x,y,z space is a ball with radius 1 and center (0,0,0). However, in
the Cartesian coordinates ρ, θ, φ the set in a) is a rectangular box.
2) [Computation and graphing of vector fields]. Given r =< x,y,z > and the
vector Field
(0.2) F(x,y,z) = F(r) =< 1 + z,yx,y >,
1
FINAL PROJECT, MATH 251, FALL 2015 2
i) Draw the arrows emanating from (x,y,z) and representing the vectors F(r) =
F(x,y,z) . First draw a 2 raw table recording F(r) versus (x,y,z) for the 4 points
(±1,±2,1) . Afterwards draw the arrows.
ii) Show that the curve
(0.3) r(t) =< x = 2cos(t), y = 4sin(t), z ≡ 0 >, 0 ≤ t < 2π,
is an ellipse. Draw the arrows emanating from (x(t),y(t),z(t)) and representing
the vector values of dr(t)
dt
, F(r(t)) = F(x(t),y(t),z(t)) . Let θ(t) be the angle
between the arrows representing dr(t)
dt
and F(r(t)) . First draw a 5 raw table
recording t, (x(t),y(t),z(t)), dr(t)
dt
, F(r(t)), cos(θ(t)) for the points (x(t),y(t),z(t))
corresponding to t = 0,π
4
, 3π
4
, 5π
4
, 7π
4
. Then draw the arrows.
iii) Given the surface
r(θ,φ) =< x = 2sin(φ)cos(θ), y = 2sin(φ)sin(θ), z = 2cos(φ) >,0 ≤ θ < 2π, 0 ≤ φ ≤ π,
in parametric form. Use trigonometric formulas to show that the following iden-
tity holds
x2(θ,φ) + y2(θ,φ) + z2(θ,φ) ≡ 22.
iv) Draw the arrows emanating from (x(θ,φ),y(θ,φ),z(θ,φ)) and representing the
vectors ∂r(θ,φ)
∂θ
× ∂r(θ,φ)
∂φ
, F(r(θ,φ)) = F(x(θ,φ),y(θ,φ),z(θ,φ)) . Let α(θ,φ) be
the angle between the arrows representing ∂r(θ,φ)
∂θ
× ∂r(θ,φ)
∂φ
and F(r(θ,φ)) . First
draw a table with raws and columns recording (θ,φ),(x(θ,φ),y ...
DOUBLE INTEGRALS PPT GTU CALCULUS (2110014)
1) The document discusses double integrals and methods for calculating them, including using iterated integrals and Riemann sums. Double integrals can represent volumes under surfaces.
2) Examples are provided to demonstrate calculating double integrals over rectangles and general regions using iterated integrals and partitioning the region.
3) There are two types of general regions: type I defined by ≤≤≤≤ and type II defined by ≤≤≤≤. The document provides methods for calculating double integrals over these region types.
double integral.pptx
1. Double integrals allow us to calculate the volume of a solid bounded above by a surface z=f(x,y) and below by a plane region R. This is done by setting up a Riemann sum of the volumes of thin rectangular prisms.
2. The limit of the Riemann sum as the number of rectangles goes to infinity gives the double integral, written as ∫∫R f(x,y) dA.
3. Example problems demonstrate calculating the double integral to find the volume in different plane regions R, including those bounded by curves such as y=x2 and y=x.
Lect1 No 873503264
This document provides an outline and definitions for fundamental concepts in set theory and discrete mathematics, including:
1. Definitions of sets, operations on sets like union and intersection, and relations.
2. Functions, relations, and properties like domains, ranges, and composition.
3. Partial orders, trees, groups, and other algebraic structures.
Notes up to_ch7_sec3
This document provides an overview of key concepts in multivariable calculus covered in MAT 10B at Williams College in Fall 2011, including:
1) Definitions and computations of double integrals, including over rectangles and more general regions, and Fubini's theorem on changing the order of integration.
2) Triple integrals as a generalization of double integrals to functions of three variables over boxes in R3.
3) Techniques for changing the order of integration in iterated integrals when one order may be more convenient than another.
Notes up to_ch7_sec3
This document provides an overview of key concepts in multivariable calculus, including:
1) Definitions and examples of double integrals, including computing integrals over rectangles using iterated integrals and properties of double integrals.
2) Computing integrals over more general regions called "elementary regions" and discussing how to integrate over the union of elementary regions.
3) Discussing how to change the order of integration when evaluating iterated integrals.
4) Providing an introduction to triple integrals, defining a "standard box" in three dimensions and computing triple integrals as iterated integrals.
Prerequisite for metric space
The document defines sets, elements, and operations on sets. It discusses that a set is a collection of objects, called elements or points. It provides examples of sets and defines subset and subset notation. The document then defines functions, including the domain, codomain, range, and image of a function. It discusses inverse functions and inverse image notation. Finally, it discusses characteristics functions, where the characteristics function of a set A maps elements in A to 1 and elements not in A to 0, completely describing the set.
24 double integral over polar coordinate
The document discusses double integrals in polar coordinates. It defines a domain D bounded by functions r1(θ) and r2(θ) between angles A and B. It partitions the domain into tiles of widths Δr and Δθ and approximates the area and volume integrals using sums over the tiles. The volume integral over the domain D of a function z=f(r,θ) is written as a double integral in polar coordinates from r1 to r2 and θ from A to B.
Relations & functions.pps
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
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The document provides an overview of random forests, including the random forest recipe, why random forests work, and ramifications of random forests. The random forest recipe involves drawing bootstrap samples to grow trees, randomly selecting features at each split, and aggregating predictions. Random forests work by decorrelating trees, which reduces variance and leads to lower prediction error compared to individual trees. Ramifications discussed include using out-of-bag samples to estimate generalization error and calculating variable importance.
Double_Integral.pdf
Here are the steps to solve this problem:
1. The region R is bounded by the circle r = 1 and the line x + y = 1 in polar coordinates.
2. To set up the double integral in polar coordinates, we first identify the limits of integration with respect to r. Since we are integrating r dr first, we hold θ fixed and let r vary from the minimum to maximum r value along each radial line. The minimum is r = 1/(cosθ + sinθ) and the maximum is r = 1.
3. Next, we identify the limits with respect to θ. The radial lines intersecting the region range from θ = 0 to θ = π/2.
4.
Density theorems for anisotropic point configurations
This document discusses density theorems for anisotropic point configurations. Specifically:
- It summarizes previous results on density theorems for linear configurations in Euclidean spaces.
- It then presents new results on density theorems for anisotropic power-type scalings, where points are scaled by different powers in different coordinates.
- Theorems are proven for anisotropic simplices and boxes in such spaces, showing that any set of positive density must contain scaled copies of these configurations for scales above a certain threshold.
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multiple intrigral lit
This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.
Introduccio al calculo vectorial
This document introduces vector fields and defines scalar and vector fields. It defines a vector field F over a planar region D as a function that assigns a vector to each point, given by the components P and Q. Similarly, a vector field over a spatial region E is given by components P, Q, and R. Examples of vector fields include velocity fields and gravitational and electric force fields. It also defines the gradient of a scalar field, divergence and curl of a vector field, and discusses integral calculations for scalar and vector fields along curves.
L1 functions, domain & range
The document defines functions and discusses:
- Functions are sets of ordered pairs with each first element paired to a unique second element.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, as equations, and as graphs.
- The domain of a function is the set of first elements and the range is the set of second elements.
International Journal of Engineering Research and Development
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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- The document discusses interval valued intuitionistic fuzzy subrings of a ring. It begins with definitions of interval valued fuzzy sets, interval valued fuzzy subrings, and interval valued intuitionistic fuzzy subsets.
- It then defines an interval valued intuitionistic fuzzy subring and establishes some properties, including that the intersection of two interval valued intuitionistic fuzzy subrings is also an interval valued intuitionistic fuzzy subring, and the intersection of a family of such subrings is also a subring.
- Proofs are provided to show that the intersection operations preserve the necessary conditions to be considered an interval valued intuitionistic fuzzy subring.
Fractional Calculus A Commutative Method on Real Analytic Functions
The document presents a method for creating a commutative fractional calculus on real analytic functions. It defines a space Rω and maps between Rω, Cω(R), and Zω such that the fractional derivative operator Dk commutes with itself. This is achieved by extending Dk from Zω to Rω and defining a map ι from Zω to Rω that preserves the properties of Dk. With these definitions, diagrams relating the fractional derivative operator to the Riemann-Liouville derivative are shown to commute for analytic functions.
Limit and continuty final
This presentation discusses limits and continuity of functions of several variables. It defines limits of multivariable functions and provides examples of limits that do and do not exist as the variables approach certain points. It also defines continuity as a function being continuous at a point if the limit of the function as the variables approach that point equals the function value at that point. Examples show functions that are and aren't continuous based on whether their limits exist or not.
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FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docx
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Density theorems for anisotropic point configurations
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1523 double integrals
- 2. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume of S = ?
- 4. Volume of ij’s column: Ayxf ijij ∆),( ** ∑∑= = ∆ m i n j ijij Ayxf 1 1 ** ),(Total volume of all columns: ij’s column: Area of Rij is Δ A = Δ x Δ y f (xij * , yij * ) Δ y Δ x x y z Rij (xi, yj) Sample point (xij * , yij * ) x y
- 5. ∑∑= = ∆≈ m i n j ijij AyxfV 1 1 ** ),( Definition ∑∑= =∞→ ∆= m i n j ijij AyxfV 1 1 ** nm, ),(lim
- 6. Definition: The double integral of f over the rectangle R is if the limit exists ∫∫R dAyxf ),( ∑∑∫∫ = =∞→ ∆= m i n j ijij R AyxfdAyxf 1 1 ** nm, ),(),( lim Double Riemann sum: ∑∑= = ∆ m i n j ijij Ayxf 1 1 ** ),(
- 7. Note 1. If f is continuous then the limit exists and the integral is defined Note 2. The definition of double integral does not depend on the choice of sample points If the sample points are upper right-hand corners then ∑∑∫∫ = =∞→ ∆= m i n j ji R AyxfdAyxf 1 1nm, ),(),( lim
- 8. Example 1 z=16-x2 -2y2 0≤x≤2 0≤y≤2 Estimate the volume of the solid above the square and below the graph
- 9. m=n=4 m=n=8 m=n=16 V≈41.5 V≈44.875 V≈46.46875 Exact volume? V=48
- 11. Integrals over arbitrary regions A R f (x,y) 0 • A is a bounded plane region • f (x,y) is defined on A • Find a rectangle R containing A • Define new function on R: ∈ = otherwise,0 ),(if),( ),( Ayxyxf yxf ∫∫∫∫ = RA dAyxfdAyxf ),(),(
- 12. Properties ∫∫∫∫∫∫ +=+ AAA dAyxgdAyxfdAyxgyxf ),(),()],(),([ ∫∫∫∫ = AA dAyxfcdAyxcf ),(),( ∫∫∫∫ ≥ AA dAyxgdAyxf ),(),( Linearity If f(x,y)≥g(x,y) for all (x,y) in R, then Comparison
- 13. ∫∫∫∫∫∫ += ∪ 2121 ),(),(),( AAAA dAyxfdAyxfdAyxf Additivity If A1 and A2 are non-overlapping regions then Area AdAdA AA ofarea1 == ∫∫∫∫ A1 A2
- 14. Computation • If f (x,y) is continuous on rectangle R=[a,b]×[c,d] then double integral is equal to iterated integral a b x y c d x y ∫∫∫∫∫∫ == b a d c d c b aR dydxyxfdxdyyxfdAyxf ),(),(),( fixed fixed
- 15. More general case • If f (x,y) is continuous on A={(x,y) | x in [a,b] and h (x) ≤ y ≤ g (x)} then double integral is equal to iterated integral a b x y h(x) g(x) x ∫ ∫∫∫ = b a xg xhA dydxyxfdAyxf )( )( ),(),( A
- 16. Similarly • If f (x,y) is continuous on A={(x,y) | y in [c,d] and h (y) ≤ x ≤ g (y)} then double integral is equal to iterated integral d x y ∫ ∫∫∫ = d c yg yhR dxdyyxfdAyxf )( )( ),(),( c h(y) g(y) y A
- 17. Note If f (x, y) = φ (x) ψ(y) then ⋅ == ∫∫∫∫∫∫ d c b a d c b aR dyydxxdxdyyxdAyxf )()()()(),( ψϕψϕ
- 18. Examples ],2/[]1,2/1[,)sin( ππ×=∫∫ AdAxyy R 2 ∫∫ − A x dAe where A is a triangle with vertices (0,0), (1,0) and (1,1)