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14.3E: Double Integrals in Polar Coordinates (Exercises)

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Terms and Concepts

1. When evaluating Rf(x,y)dA using polar coordinates, f(x,y) is replaced with _______ and dA is replaced with _______.

Answer
f(x,y) is replaced with f(rcosθ,rsinθ) and dA is replaced with rdrdθ.

2. Why would one be interested in evaluating a double integral with polar coordinates?

Defining Polar Regions

In exercises 3 - 6, express the region R in polar coordinates.

3) R is the region of the disk of radius 2 centered at the origin that lies in the first quadrant.

Answer
R={(r,θ)|0r2, 0θπ2}

4) R is the region of the disk of radius 3 centered at the origin.

5) R is the region between the circles of radius 4 and radius 5 centered at the origin that lies in the second quadrant.

Answer
R={(r,θ)|4r5, π2θπ}

6) R is the region bounded by the y-axis and x=1y2.

7) R is the region bounded by the x-axis and y=2x2.

Answer
R={(r,θ)|0r2, 0θπ}

8) R={(x,y)|x2+y24x}

9) R={(x,y)|x2+y24y}

Answer
R={(r,θ)|0r4 sinθ, 0θπ}

In exercises 10 - 15, the graph of the polar rectangular region D is given. Express D in polar coordinates.  In exercises 10 - 13 the regions are bounded by circles centered at the origin, the coordinate axes, and/or lines of the  form y=±x (each of which should be able to be identified by inspection).

10)
Half an annulus D is drawn in the first and second quadrants with inner radius 3 and outer radius 5.
11)
A sector of an annulus D is drawn between theta = pi/4 and theta = pi/2 with inner radius 3 and outer radius 5.
Answer
D={(r,θ)|3r5, π4θπ2}

12)

Half of an annulus D is drawn between theta = pi/4 and theta = 5 pi/4 with inner radius 3 and outer radius 5.

13)
A sector of an annulus D is drawn between theta = 3 pi/4 and theta = 5 pi/4 with inner radius 3 and outer radius 5.

Answer
D={(r,θ)|3r5, 3π4θ5π4}
14) In the following graph, the region D is situated below y=x and is bounded by x=1, x=5, and y=0.

A region D is given that is bounded by y = 0, x = 1, x = 5, and y = x, that is, a right triangle with a corner cut off.

15) In the following graph, the region D is bounded by y=x and y=x2.

A region D is drawn between y = x and y = x squared, which looks like a deformed lens, with the bulbous part below the straight part.

Answer
D={(r,θ)|0rtanθ secθ, 0θπ4}

Evaluating Polar Double Integrals

In exercises 16 - 25, evaluate the double integral Rf(x,y)dA over the polar rectangular region R.

16) f(x,y)=x2+y2, R={(r,θ)|3r5, 0θ2π}

17) f(x,y)=x+y, R={(r,θ)|3r5, 0θ2π}

Answer
0

18) f(x,y)=x2+xy, R={(r,θ)|1r2, πθ2π}

19) f(x,y)=x4+y4, R={(r,θ)|1r2, 3π2θ2π}

Answer
63π16

20) f(x,y)=3x2+y2, where R={(r,θ)|0r1, π2θπ}.

21) f(x,y)=x4+2x2y2+y4, where R={(r,θ)|3r4, π3θ2π3}.

Answer
3367π18

22) f(x,y)=sin(arctanyx), where R={(r,θ)|1r2, π6θπ3}

23) f(x,y)=arctan(yx), where R={(r,θ)|2r3, π4θπ3}

Answer
35π2576

24) Rex2+y2[1+2 arctan(yx)]dA, R={(r,θ)|1r2, π6θπ3}

25) R(ex2+y2+x4+2x2y2+y4)arctan(yx)dA, R={(r,θ)|1r2, π4θπ3}

Answer
7576π2(21e+e4)

Converting Double Integrals to Polar Form

In exercises 26 - 29, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

26) 21x0(x2+y2)dy dx=π402 secθsecθr3dr dθ

27) 32x0xx2+y2dy dx=π/403secθ2secθr cosθ dr dθ

Answer
52ln(1+2)

28) 10xx21x2+y2dy dx=π/40tanθ secθ0 dr dθ

29) 10xx2yx2+y2dy dx=π/40tanθ secθ0r sinθ dr dθ

Answer
16(22)

In exercises 30 - 37, draw the region of integration, R, labeling all limits of integration, convert the integrals to polar coordinates and evaluate them.

30) 309y20(x2+y2)dx dy

31) 204y24y2(x2+y2)2dx dy

Answer
π020r5dr dθ=32π3

32) 101x20(x+y) dy dx

33) 4016x216x2sin(x2+y2) dy dx

Answer
π/2π/240r sin(r2) dr dθ=π sin28

34) 5025x225x2x2+y2dydx

35) 44016y2(2yx)dxdy

Answer
3π2π240(2rsinθrcosθ)rdr dθ=1283

36) 208y2y(x+y)dxdy

37) 124x20(x+5)dydx+114x21x2(x+5)dydx+214x20(x+5)dydx

Answer
π021(rcosθ+5)rdr dθ=15π2

38) Evaluate the integral DrdA where D is the region bounded by the polar axis and the upper half of the cardioid r=1+cosθ.

39) Find the area of the region D bounded by the polar axis and the upper half of the cardioid r=1+cosθ.

Answer
3π4

40) Evaluate the integral DrdA, where D is the region bounded by the part of the four-leaved rose r=sin2θ situated in the first quadrant (see the following figure).

A region D is drawn in the first quadrant petal of the four petal rose given by r = sin (2 theta).

41) Find the total area of the region enclosed by the four-leaved rose r=sin2θ (see the figure in the previous exercise).

Answer
π2

42) Find the area of the region D which is the region bounded by y=4x2, x=3, x=2, and y=0.

43) Find the area of the region D, which is the region inside the disk x2+y24 and to the right of the line x=1.

Answer
13(4π33)

44) Determine the average value of the function f(x,y)=x2+y2 over the region D bounded by the polar curve r=cos2θ, where π4θπ4 (see the following graph).

The first/fourth-quadrant petal of the four-petal rose given by r = cos (2 theta) is shown.

45) Determine the average value of the function f(x,y)=x2+y2 over the region D bounded by the polar curve r=3sin2θ, where 0θπ2 (see the following graph).

The first-quadrant petal of the four-petal rose given by r = 3sin (2 theta) is shown.

Answer
163π

46) Find the volume of the solid situated in the first octant and bounded by the paraboloid z=14x24y2 and the planes x=0, y=0, and z=0.

47) Find the volume of the solid bounded by the paraboloid z=29x29y2 and the plane z=1.

Answer
π18

48)

  1. Find the volume of the solid S1 bounded by the cylinder x2+y2=1 and the planes z=0 and z=1.
  2. Find the volume of the solid S2 outside the double cone z2=x2+y2 inside the cylinder x2+y2=1, and above the plane z=0.
  3. Find the volume of the solid inside the cone z2=x2+y2 and below the plane z=1 by subtracting the volumes of the solids S1 and S2.

49)

  1. Find the volume of the solid S1 inside the unit sphere x2+y2+z2=1 and above the plane z=0.
  2. Find the volume of the solid S2 inside the double cone (z1)2=x2+y2 and above the plane z=0.
  3. Find the volume of the solid outside the double cone (z1)2=x2+y2 and inside the sphere x2+y2+z2=1.
Answer
a. 2π3; b. π2; c. π6

In Exercises 50-51, special double integrals are presented that are especially well suited for evaluation in polar coordinates.

50) The surface of a right circular cone with height h and base radius a can be described by the equation f(x,y)=hhx2a2+y2a2, where the tip of the cone lies at (0,0,h) and the circular base lies in the xy-plane, centered at the origin.
Confirm that the volume of a right circular cone with height h and base radius a is V=13πa2h by evaluating Rf(x,y)dA in polar coordinates.

51) Consider Re(x2+y2)dA.
(a) Why is this integral difficult to evaluate in rectangular coordinates, regardless of the region R?
(b) Let R be the region bounded by the circle of radius a centered at the origin. Evaluate the double integral using polar coordinates.
(c) Take the limit of your answer from (b), as a. What does this imply about the volume under the surface of e(x2+y2) over the entire xy-plane?

For the following two exercises, consider a spherical ring, which is a sphere with a cylindrical hole cut so that the axis of the cylinder passes through the center of the sphere (see the following figure).

A spherical ring is shown, that is, a sphere with a cylindrical hole going all the way through it.

52) If the sphere has radius 4 and the cylinder has radius 2 find the volume of the spherical ring.

53) A cylindrical hole of diameter 6 cm is bored through a sphere of radius 5 cm such that the axis of the cylinder passes through the center of the sphere. Find the volume of the resulting spherical ring.

Answer
256π3 cm3

54) Find the volume of the solid that lies under the double cone z2=4x2+4y2, inside the cylinder x2+y2=x, and above the plane z=0.

55) Find the volume of the solid that lies under the paraboloid z=x2+y2, inside the cylinder x2+y2=1 and above the plane z=0.

Answer
3π32

56) Find the volume of the solid that lies under the plane x+y+z=10 and above the disk x2+y2=4x.

57) Find the volume of the solid that lies under the plane 2x+y+2z=8 and above the unit disk x2+y2=1.

Answer
4π

58) A radial function f is a function whose value at each point depends only on the distance between that point and the origin of the system of coordinates; that is, f(x,y)=g(r), where r=x2+y2. Show that if f is a continuous radial function, then

Df(x,y)dA=(θ2θ1)[G(R2)G(R1)], where G(r)=rg(r) and (x,y)D={(r,θ)|R1rR2, 0θ2π}, with 0R1<R2 and 0θ1<θ22π.

59) Use the information from the preceding exercise to calculate the integral D(x2+y2)3dA, where D is the unit disk.

Answer
π4

60) Let f(x,y)=F(r)r be a continuous radial function defined on the annular region D={(r,θ)|R1rR2, 0θ2π}, where r=x2+y2, 0<R1<R2, and F is a differentiable function.

Show that Df(x,y)dA=2π[F(R2)F(R1)].

61) Apply the preceding exercise to calculate the integral Dex2+y2x2+y2dx dy where D is the annular region between the circles of radii 1 and 2 situated in the third quadrant.

Answer
12πe(e1)

62) Let f be a continuous function that can be expressed in polar coordinates as a function of θ only; that is, f(x,y)=h(θ), where (x,y)D={(r,θ)|R1rR2, θ1θθ2}, with 0R1<R2 and 0θ1<θ22π.

Show that Df(x,y)dA=12(R22R21)[H(θ2)H(θ1)], where H is an antiderivative of h.

63) Apply the preceding exercise to calculate the integral Dy2x2dA, where D={(r,θ)|1r2, π6θπ3}.

Answer
3π4

64) Let f be a continuous function that can be expressed in polar coordinates as a function of θ only; that is f(x,y)=g(r)h(θ), where (x,y){(r,θ)|R1rR2, θ1θθ2} with 0R1<R2 and 0θ1<θ22π. Show that Df(x,y)dA=[G(R2)G(R1)] [H(θ2)H(θ1)], where G and H are antiderivatives of g and h, respectively.

65) Evaluate Darctan(yx)x2+y2dA, where D={(r,θ)|2r3, π4θπ3}.

Answer
133864π2

66) A spherical cap is the region of a sphere that lies above or below a given plane.

a. Show that the volume of the spherical cap in the figure below is 16πh(3a2+h2).

A sphere of radius R has a circle inside of it h units from the top of the sphere. This circle has radius a, which is less than R.

b. A spherical segment is the solid defined by intersecting a sphere with two parallel planes. If the distance between the planes is h show that the volume of the spherical segment in the figure below is 16πh(3a2+3b2+h2).

A sphere has two parallel circles inside of it h units apart. The upper circle has radius b, and the lower circle has radius a. Note that a > b.

67) In statistics, the joint density for two independent, normally distributed events with a mean μ=0 and a standard distribution σ is defined by p(x,y)=12πσ2ex2+y22σ2. Consider (X,Y), the Cartesian coordinates of a ball in the resting position after it was released from a position on the z-axis toward the xy-plane. Assume that the coordinates of the ball are independently normally distributed with a mean μ=0 and a standard deviation of σ (in feet). The probability that the ball will stop no more than a feet from the origin is given by P[X2+Y2a2]=Dp(x,y)dy dx, where D is the disk of radius a centered at the origin. Show that P[X2+Y2a2]=1ea2/2σ2.

68) The double improper integral ex2+y2/2dydx may be defined as the limit value of the double integrals Dex2+y2/2dA over disks Da of radii a centered at the origin, as a increases without bound; that is,

ex2+y2/2dy dx=lim

Use polar coordinates to show that \displaystyle \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \, dx = 2\pi.

69) Show that \displaystyle \int_{-\infty}^{\infty} e^{-x^2/2}\,dx = \sqrt{2\pi} by using the relation

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \,dx = \left(\int_{-\infty}^{\infty} e^{-x^2/2}dx \right) \left( \int_{-\infty}^{\infty} e^{-y^2/2}dy \right). \nonumber

Contributors

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

  • Problems 1, 2, 34 - 37 and 50 - 51 are from Apex Calculus, Chapter 13.3
  • Edited by Paul Seeburger (Monroe Community College)

 


14.3E: Double Integrals in Polar Coordinates (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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