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3.7.E: Change of Variables in Definite Integrals (Exercises)

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Exercise 3.7.E.1

Find the area of the region enclosed by the ellipse with equation x2+4y2=4.

Answer

2π

Exercise 3.7.E.2

Given a>0 and b>0, show that the area enclosed by the ellipse with equation

x2a2+y2b2=1

is πab.

Exercise 3.7.E.3

Find the volume of the region enclosed by the ellipsoid with equation

x225+y2+z24=1.

Answer

40π3

Exercise 3.7.E.4

Given a>0, b>0, and c>0, show that the volume of the region enclosed by the ellipsoid

x2a2+y2b2+z2c2=1

is 43πabc.

Exercise 3.7.E.5

Find the polar coordinates for each of the following points given in Cartesian coordinates.

(a) (1,1)

(b) (-2,3)

(c) (-1,3)

(d) (4,-4)

Answer

(a) (2,π4)

(c) (10,4.3906)

Exercise 3.7.E.6

Find the Cartesian coordinates for each of the following points given in polar coordinates.

(a) (3,0)

(b) (2,5π6)

(c) (5,π)

(d) (4,4π3)

Answer

(a) (3,0)

(c) (-5,0)

Exercise 3.7.E.7

Evaluate

D(x2+y2)dxdy,

where D is the disk in R2 of radius 2 centered at the origin.

Answer

D(x2+y2)dxdy=8π

Exercise 3.7.E.8

Evaluate

Dsin(x2+y2)dxdy,

where D is the disk in R2 of radius 1 centered at the origin.

Exercise 3.7.E.9

Evaluate

D1x2+y2dxdy,

where D is the region in the first quadrant of R2 which lies between the circle with equation x2+y2=1 and the circle with equation x2+y2=16.

Answer

D1x2+y2dxdy=πlog(2)

Exercise 3.7.E.10

Evaluate

Dlog(x2+y2)dxdy,

where D is the region in R2 which lies between the circle with equation x2+y2=1 and the circle with equation x2+y2=4.

Answer

Dlog(x2+y2)dxdy=π(8log(2)3)

Exercise 3.7.E.11

Using polar coordinates, verify that the area of a circle of radius r is πr2.

Exercise 3.7.E.12

Let

I=ex22dx.

(a) Show that

I2=e12(x2+y2)dxdy.

(b) Show that

I2=02π0rer22dθdr.

(c) Show that

ex22dx=2π.

Exercise 3.7.E.13

Find the spherical coordinates of the point with Cartesian coordinates (1,1,2).

Answer

(6,3π4,0.6155)

Exercise 3.7.E.14

Find the spherical coordinates of the point with Cartesian coordinates (3,2,1).

Exercise 3.7.E.15

Find the Cartesian coordinates of the point with spherical coordinates (2,3π4,2π3).

Answer

(32,32,1)

Exercise 3.7.E.16

Find the Cartesian coordinates of the point with spherical coordinates (5,5π3,π6).

Exercise 3.7.E.17

Evaluate

(x2+y2+z2)dxdydz,

where D is the closed ball in R3 of radius 2 centered at the origin.

Answer

D(x2+y2+z2)dxdydz=128π5

Exercise 3.7.E.18

Evaluate

D1x2+y2+z2dxdydz,

where D is the region in R3 between the two spheres with equations x2+y2+z2=4 and x2+y2+z2=9.

Exercise 3.7.E.19

Evaluate

Dsin(x2+y2+z2)dxdydz,

where D is the region in R3 described by x0,y0,z0, and x2+y2+z21.

Answer

Dsin(x2+y2+z2dxdydz=π2(2sin(1)+cos(1)2)0.3506

Exercise 3.7.E.20

Evaluate

De(x2+y2+z2)dxdydz,

where D is the closed ball in R3 of radius 3 centered at the origin.

Exercise 3.7.E.21

Let D be the region in R3 described by x2+y2+z21 and zx2+y2.

(a) Explain why the spherical coordinate change of variables maps the region

E={(ρ,θ,φ):0ρ1,0θ2π,0φπ4}

onto D.

(b) Find the volume of D.

Answer

(b) π3(22)

Exercise 3.7.E.22

If a point P has Cartesian coordinates (x,y,z), then the cylindrical coordinates of P are (r,θ,z), where r and θ are the polar coordinates of (x,y). Show that

|det(x,y,z)(r,θ,z)|=r.

Exercise 3.7.E.23

Use cylindrical coordinates to evaluate

Dx2+y2dxdydz,

where D is the region in R3 described by 1x2+y24 and 0z5.

Answer

Dx2+y2dxdydz=70π3

Exercise 3.7.E.24

A drill with a bit with a radius of 1 centimeter is used to drill a hole through the center of a solid ball of radius 3 centimeters. What is the volume of the remaining solid?

Exercise 3.7.E.25

Let D be the set of all points in the intersection of the two solid cylinders in R3 described by x2+y21 and x2+z21. Find the volume of D.

Answer

163


This page titled 3.7.E: Change of Variables in Definite Integrals (Exercises) is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform.

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