Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

7.2E: Exercises

  • Page ID
    25941
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Exercises \(\PageIndex{1-14}\)

    In the following exercises, specify whether the region is of Type I or Type II.

    1. The region \(D\) bounded by \(y = x^3, \space y = x^3 + 1, \space x = 0,\) and \(x = 1\) as given in the following figure.

    A region is bounded by y = 1 + x cubed, y = x cubed, x = 0, and x = 1.

    2. Find the average value of the function \(f(x,y) = 3xy\) on the region graphed in the previous exercise.

    Answer

    \(\frac{27}{20}\)

    3. Find the area of the region \(D\) given in the previous exercise.

    4. The region \(D\) bounded by \(y = sin \space x, \space y = 1 + sin \space x, \space x = 0\), and \(x = \frac{\pi}{2}\) as given in the following figure.

    A region is bounded by y = 1 + sin x, y = sin x, x = 0, and x = pi/2.

    Answer

    Type I but not Type II

    5. Find the average value of the function \(f(x,y) = cos \space x\) on the region graphed in the previous exercise.

    6. Find the area of the region \(D\) given in the previous exercise.

    Answer

    \(\frac{\pi}{2}\)

    7. The region \(D\) bounded by \(x = y^2 - 1\) and \(x = \sqrt{1 - y^2}\) as given in the following figure.

    A region is bounded by x = negative 1 + y squared and x = the square root of the quantity (1 minus y squared).

    8. Find the volume of the solid under the graph of the function \(f(x,y) = xy + 1\) and above the region in the figure in the previous exercise.

    Answer

    \(\frac{1}{6}(8 + 3\pi)\)

    9. The region \(D\) bounded by \(y = 0, \space x = -10 + y,\) and \(x = 10 - y\) as given in the following figure.

    A region is bounded by x = negative 10 + y, x = 10 minus y, and y = 0.

    10. Find the volume of the solid under the graph of the function \(f(x,y) = x + y\) and above the region in the figure from the previous exercise.

    Answer

    \(\frac{1000}{3}\)

    11. The region \(D\) bounded by \(y = 0, \space x = y - 1, \space x = \frac{\pi}{2}\) as given in the following figure.

    A region is bounded by x = pi/2, y = 0, and x = negative 1 + y.

    12. The region \(D\) bounded by \(y = 0\) and \(y = x^2 - 1\) as given in the following figure.

    A region is bounded by y = 0 and y = negative 1 + x squared.

    Answer

    Type I and Type II

    13. Let \(D\) be the region bounded by the curves of equations \(y = x, \space y = -x\) and \(y = 2 - x^2\). Explain why \(D\) is neither of Type I nor II.

    14. Let \(D\) be the region bounded by the curves of equations \(y = cos \space x\) and \(y = 4 - x^2\) and the \(x\)-axis. Explain why \(D\) is neither of Type I nor II.

    Answer

    The region \(D\) is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions \(g_1(x)\) and \(g_2(x)\). The region is not of Type II: it does not lie between two horizontal lines and the graphs of two continuous functions \(h_1(y)\) and \(h_2(y)\).

    Exercises \(\PageIndex{15-20}\)

    In the following exercises, evaluate the double integral \(\iint_D f(x,y) dA\) over the region \(D\).

    15. \(f(x,y) = 2x + 4y\) and

    \(D = \big\{(x,y)|\, 0 \leq x \leq 1, \space x^3 \leq y \leq x^3 + 1 \big\}\)

    16. \(f(x,y) = 1\) and

    \(D = \big\{(x,y)| \, 0 \leq x \leq \frac{\pi}{2}, \space \sin x \leq y \leq 1 + \sin x \big\}\)

    Answer

    \(\frac{\pi}{2}\)

    17. \(f(x,y) = 2\) and

    \(D = \big\{(x,y)| \, 0 \leq y \leq 1, \space y - 1 \leq x \leq \arccos y \big\}\)

    18. \(f(x,y) = xy\) and

    \(D = \big\{(x,y)| \, -1 \leq y \leq 1, \space y^2 - 1 \leq x \leq \sqrt{1 - y^2} \big\}\)

    Answer

    0

    19. \(f(x,y) = sin \space y\) and \(D\) is the triangular region with vertices \((0,0), \space (0,3)\), and \((3,0)\)

    20. \(f(x,y) = -x + 1\) and \(D\) is the triangular region with vertices \((0,0), \space (0,2)\), and \((2,2)\)

    Answer

    \(\frac{2}{3}\)

    Exercises \(\PageIndex{21-26}\)

    Evaluate the iterated integrals.

    21. \[\int_0^3 \int_{2x}^{3x} (x + y^2) dy \space dx\]

    22. \[\int_0^1 \int_{2\sqrt{x}}^{2\sqrt{x}+1} (xy + 1) dy \space dx\]

    Answer

    \(\frac{41}{20}\)

    23. \[\int_e^{e^2} \int_{ln \space u}^2 (v + ln \space u) dv \space du\]

    24. \[\int_1^2 \int_{-u^2-1}^{-u} (8 uv) dv \space du\]

    Answer

    \(-63\)

    25. \[\int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} (2x + 4y^3) dx \space dy\]

    26. \[\int_0^1 \int_{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}} 4dx \space dy\]

    Answer

    \(\pi\)

    Exercises \(\PageIndex{27-29}\)

    27. Let \(D\) be the region bounded by \(y = 1 - x^2, \space y = 4 - x^2\), and the \(x\)- and \(y\)-axes.

    a. Show that

    \[\iint_D xdA = \int_0^1 \int_{1-x^2}^{4-x^2} x \space dy \space dx + \int_1^2 \int_0^{4-x^2} x \space dy \space dx\] by dividing the region \(D\) into two regions of Type I.

    b. Evaluate the integral \[\iint_D s \space dA.\]

    28. Let \(D\) be the region bounded by \(y = 1, \space y = x, \space y = ln \space x\), and the \(x\)-axis.

    a. Show that

    \[\iint_D y^2 dA = \int_{-1}^0 \int_{-x}^{2-x^2} y^2 dy \space dx + \int_0^1 \int_x^{2-x^2} y^2 dy \space dx\] by dividing the region \(D\) into two regions of Type I, where \(D = {(x,y)|y \geq x, y \geq -x, \space y \leq 2-x^2}\).

    b. Evaluate the integral \[\iint_D y^2 dA.\]

    29. Let \(D\) be the region bounded by \(y = x^2\), \(y = x + 2\), and \(y = -x\).

    a. Show that \[\iint_D x \space dA = \int_0^1 \int_{-y}^{\sqrt{y}} x \space dx \space dy + \int_1^2 \int_{y-2}^{\sqrt{y}} x \space dx \space dy\] by dividing the region \(D\) into two regions of Type II, where \(D = {(x,y)|y \geq x^2, \space y \geq -x, \space y \leq x + 2}\).

    b. Evaluate the integral \[\iint_D x \space dA.\]

    Answer

    a. Answers may vary; b. \(\frac{8}{12}\)

    Exercises \(\PageIndex{30-33}\)

    30. The region \(D\) bounded by \(x = 0, y = x^5 + 1\), and \(y = 3 - x^2\) is shown in the following figure. Find the area \(A(D)\) of the region \(D\).

    A region is bounded by y = 1 + x to the fifth power, y = 3 minus x squared, and x = 0.

    31. The region \(D\) bounded by \(y = cos \space x, \space y = 4 \space cos \space x\), and \(x = \pm \frac{\pi}{3}\) is shown in the following figure. Find the area \(A(D)\) of the region \(D\).

    A region is bounded by y = cos x, y = 4 + cos x, x = negative 1, and x = 1.

    Answer

    \(\frac{8\pi}{3}\)

    32. Find the area \(A(D)\) of the region \(D = \big\{(x,y)| \, y \geq 1 - x^2, y \leq 4 - x^2, \space y \geq 0, \space x \geq 0 \big\}\).

    33. Let \(D\) be the region bounded by \( y = 1, \space y = x, \space y = ln \space x\), and the \(x\)-axis. Find the area \(A(D)\) of the region \(D\).

    Answer

    \(e - \frac{3}{2}\)

    Exercises \(\PageIndex{34-35}\)

    34. Find the average value of the function \(f(x,y) = sin \space y\) on the triangular region with vertices \((0,0), \space (0,3)\), and \((3,0)\).

    35. Find the average value of the function \(f(x,y) = -x + 1\) on the triangular region with vertices \((0,0), \space (0,2)\), and \((2,2)\).

    Answer

    \(\frac{2}{3}\)

    Exercises \(\PageIndex{36-39}\)

    In the following exercises, change the order of integration and evaluate the integral.

    36. \[\int_{-1}^{\pi/2} \int_0^{x+1} sin \space x \space dy \space dx\]

    37. \[\int_0^1 \int_{x-1}^{1-x} x \space dy \space dx\]

    Answer

    \[\int_0^1 \int_{x-1}^{1-x} x \space dy \space dx = \int_{-1}^0 \int_0^{y+1} x \space dx \space dy + \int_0^1 \int_-^{1-y} x \space dx \space dy = \frac{1}{3}\]

    38. \[\int_{-1}^0 \int_{-\sqrt{y+1}}^{\sqrt{y+1}} y^2 dx \space dy\]

    39. \[\int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy\]

    Answer

    \[\int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy = \int_1^2 \int_{-\sqrt{x^2-1}}^{\sqrt{x^2-1}} y \space dy \space dx = 0\]

    Exercises \(\PageIndex{40-41}\)

    40. The region \(D\) is shown in the following figure. Evaluate the double integral \[\iint_D (x^2 + y) dA\] by using the easier order of integration.

    A region is bounded by y = negative 4 + x squared and y = 4 minus x squared.

    41. The region \(D\) is shown in the following figure. Evaluate the double integral \[\iint_D (x^2 - y^2) dA\] by using the easier order of integration.

    A region is bounded by y to the fourth power = 1 minus x and y to the fourth power = 1 + x.

    Answer

    \[\iint_D (x^2 - y^2) dA = \int_{-1}^1 \int_{y^4-1}^{1-y^4} (x^2 - y^2)dx \space dy = \frac{464}{4095}\]

    Exercises \(\PageIndex{42-45}\)

    42. Find the volume of the solid under the surface \(z = 2x + y^2\) and above the region bounded by \(y = x^5\) and \(y = x\).

    43. Find the volume of the solid under the plane \(z = 3x + y\) and above the region determined by \(y = x^7\) and \(y = x\).

    Answer

    \(\frac{4}{5}\)

    44. Find the volume of the solid under the plane \(z = 3x + y\) and above the region bounded by \(x = tan \space y, \space x = -tan \space y\), and \(x = 1\).

    45. Find the volume of the solid under the surface \(z = x^3\) and above the plane region bounded by \(x = sin \space y, \space x = -sin \space y\), and \(x = 1\).

    Answer

    \(\frac{5\pi}{32}\)

    Exercises \(\PageIndex{46-47}\)

    46. Let \(g\) be a positive, increasing, and differentiable function on the interval \([a,b]\). Show that the volume of the solid under the surface \(z = g'(x)\) and above the region bounded by \(y = 0, \space y = g(x), \space x = a\), and \(x = b\) is given by \(\frac{1}{2}(g^2 (b) - g^2 (a))\).

    47. Let \(g\) be a positive, increasing, and differentiable function on the interval \([a,b]\) and let \(k\) be a positive real number. Show that the volume of the solid under the surface \(z = g'(x)\) and above the region bounded by \(y = g(x), \space y = g(x) + k, \space x = a\), and \(x = b\) is given by \(k(g(b) - g(a)).\)

    Exercises \(\PageIndex{48-51}\)

    48. Find the volume of the solid situated in the first octant and determined by the planes \(z = 2\), \(z = 0, \space x + y = 1, \space x = 0\), and \(y = 0\).

    49. Find the volume of the solid situated in the first octant and bounded by the planes \(x + 2y = 1\), \(x = 0, \space z = 4\), and \(z = 0\).

    Answer

    1

    50. Find the volume of the solid bounded by the planes \(x + y = 1, \space x - y = 1, \space x = 0, \space z = 0\), and \(z = 10\).

    51. Find the volume of the solid bounded by the planes \(x + y = 1, \space x - y = 1, \space x - y = -1, \space z = 1\), and \(z = 0\)

    Answer

    2

    Exercises \(\PageIndex{52}\)

    52. Let \(S_1\) and \(S_2\) be the solids situated in the first octant under the planes \(x + y + z = 1\) and \(x + y + 2z = 1\) respectively, and let \(S\) be the solid situated between \(S_1, \space S_2, \space x = 0\), and \(y = 0\).

    1. Find the volume of the solid \(S_1\).
    2. Find the volume of the solid \(S_2\).
    3. Find the volume of the solid \(S\) by subtracting the volumes of the solids \(S_1\) and \(S_2\).

    53. Let \(S_1\) and \(S_2\) be the solids situated in the first octant under the planes \(2x + 2y + z = 2\) and \(x + y + z = 1\) respectively, and let \(S\) be the solid situated between \(S_1, \space S_2, \space x = 0\), and \(y = 0\).

    1. Find the volume of the solid \(S_1\).
    2. Find the volume of the solid \(S_2\).
    3. Find the volume of the solid \(S\) by subtracting the volumes of the solids \(S_1\) and \(S_2\).
    Answer

    a. \(\frac{1}{3}\); b. \(\frac{1}{6}\); c. \(\frac{1}{6}\)

    54. Let \(S_1\) and \(S_2\) be the solids situated in the first octant under the plane \(x + y + z = 2\) and under the sphere \(x^2 + y^2 + z^2 = 4\), respectively. If the volume of the solid \(S_2\) is \(\frac{4\pi}{3}\) determine the volume of the solid \(S\) situated between \(S_1\) and \(S_2\) by subtracting the volumes of these solids.

    55. Let \(S_1\) and \(S_2\) be the solids situated in the first octant under the plane \(x + y + z = 2\) and under the sphere \(x^2 + y^2 = 4\), respectively.

    1. Find the volume of the solid \(S_1\).
    2. Find the volume of the solid \(S_2\).
    3. Find the volume of the solid \(S\) situated between \(S_1\) and \(S_2\) by subtracting the volumes of the solids \(S_1\) and \(S_2\).
    Answer

    a. \(\frac{4}{3}\); b. \(2\pi\); c. \(\frac{6\pi - 4}{3}\)

    Exercises \(\PageIndex{56-57}\)

    56. [T] The following figure shows the region \(D\) bounded by the curves \(y = sin \space x, \space x = 0\), and \(y = x^4\). Use a graphing calculator or CAS to find the