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=∫∞−∞e−x22dx.
(a) Show that
I2=∫∞−∞∫∞−∞e−12(x2+y2)dxdy.
(b) Show that
I2=∫∞0∫2π0re−r22dθdr.
(c) Show that
∫∞−∞e−x22dx=√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
∭D1√x2+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 x≥0,y≥0,z≥0, and x2+y2+z2≤1.
- 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+z2≤1 and z≥√x2+y2.
(a) Explain why the spherical coordinate change of variables maps the region
E={(ρ,θ,φ):0≤ρ≤1,0≤θ≤2π,0≤φ≤π4}
onto D.
(b) Find the volume of D.
- Answer
-
(b) π3(2−√2)
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
∬D√x2+y2dxdydz,
where D is the region in R3 described by 1≤x2+y2≤4 and 0≤z≤5.
- Answer
-
∭D√x2+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+y2≤1 and x2+z2≤1. Find the volume of D.
- Answer
-
163