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Mathematics LibreTexts

7.1E: Exercises

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    Exercise \(\PageIndex{1-2}\)

    In the following exercises, use the midpoint rule with \(m = 4\) and \(n = 2\) to estimate the volume of the solid bounded by the surface \(z = f(x,y)\), the vertical planes \(x = 1\), \(x = 2\), \(y = 1\), and \(y = 2\), and the horizontal plane \(x = 0\).

    1. \(f(x,y) = 4x + 2y + 8xy\)

    Answer:

    27

     

    2. \(f(x,y) = 16x^2 + \frac{y}{2}\)

    Exercise \(\PageIndex{3-4}\)

    In the following exercises, estimate the volume of the solid under the surface \(z = f(x,y)\) and above the rectangular region R by using a Riemann sum with \(m = n = 2\) and the sample points to be the lower left corners of the subrectangles of the partition.

    3. \(f(x,y) = sin \space x - cos \space y\), \(R = [0, \pi] \times [0, \pi]\)

    Answer:

    0

    4. \(f(x,y) = cos \space x + cos \space y\), \(R = [0, \pi] \times [0, \frac{\pi}{2}]\)

    Exercise \(\PageIndex{5-8}\)

     5. Use the midpoint rule with \(m = n = 2\) to estimate \(\iint_R f(x,y) dA\), where the values of the function f on \(R = [8,10] \times [9,11]\) are given in the following table.

      y
    x 9 9.5 10 10.5 11
    8 9.8 5 6.7 5 5.6
    8.5 9.4 4.5 8 5.4 3.4
    9 8.7 4.6 6 5.5 3.4
    9.5 6.7 6 4.5 5.4 6.7
    10 6.8 6.4 5.5 5.7 6.8
    Answer:

    21.3.

     

    6. The values of the function f on the rectangle \(R = [0,2] \times [7,9]\) are given in the following table. Estimate the double integral \(\iint_R f(x,y)dA\) by using a Riemann sum with \(m = n = 2\). Select the sample points to be the upper right corners of the subsquares of R.

      \(y_0 = 7\) \(y_1 = 8\) \(y_2 = 9\)
    \(x_0 = 0\) 10.22 10.21 9.85
    \(x_1 = 1\) 6.73 9.75 9.63
    \(x_2 = 2\) 5.62 7.83 8.21

     7. The depth of a children’s 4-ft by 4-ft swimming pool, measured at 1-ft intervals, is given in the following table.

    1. Estimate the volume of water in the swimming pool by using a Riemann sum with \(m = n = 2\). Select the sample points using the midpoint rule on \(R = [0,4] \times [0,4]\).
    2. Find the average depth of the swimming pool.

       

        y
      x 0 1 2 3 4
      0 1 1.5 2 2.5 3
      1 1 1.5 2 2.5 3
      2 1 1.5 1.5 2.5 3
      3 1 1 1.5 2 2.5
      4 1 1 1 1.5 2
    Answer:

    a. 28 \(ft^3\) b. 1.75 ft.

    8. The depth of a 3-ft by 3-ft hole in the ground, measured at 1-ft intervals, is given in the following table.

    1. Estimate the volume of the hole by using a Riemann sum with \(m = n = 3\) and the sample points to be the upper left corners of the subsquares of \(R\).
    2. Find the average depth of the hole.
        y
      x 0 1 2 3
      0 6 6.5 6.4 6
      1 6.5 7 7.5 6.5
      2 6.5 6.7 6.5 6
      3 6 6.5 5 5.6
     

    Exercise \(\PageIndex{9-10}\)

    9. The level curves \(f(X,Y) = K\) of the function f are given in the following graph, where k is a constant.

    1. Apply the midpoint rule with \(M = N = 2\)m=n=2m=n=2 to estimate the double integral \(\iint_R f(x,y)dA\), where \(R = [0.2,1] \times [0,0.8]\).
    2. Estimate the average value of the function f on \(R\).

      A series of curves marked k = negative 1, negative ½, negative ¼, negative 1/8, 0, 1/8, ¼, ½, and 1. The line marked k = 0 serves as an asymptote along the line y = x. The lines originate at (along the y axis) 1, 0.7, 0.5, 0.38, 0, (along the x axis) 0.38, 0.5, 0.7, and 1, with the further out lines curving less dramatically toward the asymptote.

    Answer:

    a. 0.112 b. \(f_{ave} ≃ 0.175\); here \(f(0.4,0.2) ≃ 0.1\), \(f(0.2,0.6) ≃− 0.2\), \(f(0.8,0.2) ≃ 0.6\), and \(f(0.8,0.6) ≃ 0.2\).

     

    10. The level curves \(f(x,y) = k\) of the function f are given in the following graph, where k is a constant.

    1. Apply the midpoint rule with \(m = n = 2\) to estimate the double integral \(\iint_R f(x,y)dA\), where \(R = [0.1,0.5] \times [0.1,0.5]\).
    2. Estimate the average value of the function f on \(R\).

      A series of quarter circles drawn in the first quadrant marked k = 1/32, 1/16, 1/8, ¼, ½, ¾, and 1. The quarter circles have radii 0. 17, 0.25, 0.35, 0.5, 0.71, 0.87, and 1, respectively.

    Exercise \(\PageIndex{11-12}\)

    11. The solid lying under the surface \(z = \sqrt{4 - y^2}\) and above the rectangular region\( R = [0,2] \times [0,2]\) is illustrated in the following graph. Evaluate the double integral \(\iint_Rf(x,y)\), where \(f(x,y) = \sqrt{4 - y^2}\) by finding the volume of the corresponding solid.

    A quarter cylinder with center along the x axis and with radius 2. It has height 2 as shown.

    Answer:

    \(2\pi\)

     

    12. The solid lying under the plane \(z = y + 4\) and above the rectangular region \(R = [0,2] \times [0,4]\) is illustrated in the following graph. Evaluate the double integral \(\iint_R f(x,y)dA\), where \(f(x,y) = y + 4\), by finding the volume of the corresponding solid.

    In xyz space, a shape is created with sides given by y = 0, x = 0, y = 4, x = 2, z = 0, and the plane the runs from z = 4 along the y axis to z = 8 along the plane formed by y = 4.

     

     

    Exercise \(\PageIndex{13-20}\)

    In the following exercises, calculate the integrals by interchanging the order of integration.

     13. \[\int_{-1}^1\left(\int_{-2}^2 (2x + 3y + 5)dx \right) \space dy\]

    Answer:

    40

    14. \[\int_0^2\left(\int_0^1 (x + 2e^y + 3)dx \right) \space dy\]

    15. \[\int_1^{27}\left(\int_1^2 (\sqrt[3]{x} + \sqrt[3]{y})dy \right) \space dx\]

    Answer:

    \(\frac{81}{2} + 39\sqrt[3]{2}\).

    16. \[\int_1^{16}\left(\int_1^8 (\sqrt[4]{x} + 2\sqrt[3]{y})dy \right) \space dx\]

    17. \[\int_{ln \space 2}^{ln \space 3}\left(\int_0^1 e^{x+y}dy \right) \space dx\]

    Answer:

    \(e - 1\).

    18. \[\int_0^2\left(\int_0^1 3^{x+y}dy \right) \space dx\]

    19. \[\int_1^6\left(\int_2^9 \frac{\sqrt{y}}{y^2}dy \right) \space dx\]

    Answer:

    \(15 - \frac{10\sqrt{2}}{9}\).

    20. \[\int_1^9 \left(\int_4^2 \frac{\sqrt{x}}{y^2}dy \right) dx\]

     

     

    Exercise \(\PageIndex{21-34}\)

    In the following exercises, evaluate the iterated integrals by choosing the order of integration.

     21. \[\int_0^{\pi} \int_0^{\pi/2} sin(2x)cos(3y)dx \space dy\]

    Answer:

    0.

    22. \[\int_{\pi/12}^{\pi/8}\int_{\pi/4}^{\pi/3} [cot \space x + tan(2y)]dx \space dy\]

    23. \[\int_1^e \int_1^e \left[\frac{1}{x}sin(ln \space x) + \frac{1}{y}cos (ln \space y)\right] dx \space dy\]

    Answer:

    \((e − 1)(1 + sin1 − cos1)\)

    24. \[\int_1^e \int_1^e \frac{sin(ln \space x)cos (ln \space y)}{xy} dx \space dy\]

    25. \[\int_1^2 \int_1^2 \left(\frac{ln \space y}{x} + \frac{x}{2y + 1}\right) dy \space dx\]

    Answer:

    \(\frac{3}{4}ln \left(\frac{5}{3}\right) + 2b \space ln^2 2 - ln \space 2\)

    26. \[\int_1^e \int_1^2 x^2 ln(x) dy \space dx\]

     

    27. \[\int_1^{\sqrt{3}} \int_1^2 y \space arctan \left(\frac{1}{x}\right) dy \space dx\]

    Answer:

    \(\frac{1}{8}[(2\sqrt{3} - 3) \pi + 6 \space ln \space 2]\).

    28. \[\int_0^1 \int_0^{1/2} (arcsin \space x + arcsin \space y) dy \space dx\]

    29. \[\int_0^1 \int_0^2 xe^{x+4y} dy \space dx\]

    Answer:

    \(\frac{1}{4}e^4 (e^4 - 1)\).

    30. \[\int_1^2 \int_0^1 xe^{x-y} dy \space dx\]

    31. \[\int_1^e \int_1^e \left(\frac{ln \space y}{\sqrt{y}} + \frac{ln \space x}{\sqrt{x}}\right) dy \space dx\]

    Answer:

    \(4(e - 1)(2 - \sqrt{e})\).

    32. \[\int_1^e \int_1^e \left(\frac{x \space ln \space y}{\sqrt{y}} + \frac{y \space ln \space x}{\sqrt{x}}\right) dy \space dx\]

    33. \[\int_0^1 \int_1^2 \left(\frac{x}{x^2 + y^2} \right) dy \space dx\]

    Answer:

    \(-\frac{\pi}{4} + ln \left(\frac{5}{4}\right) - \frac{1}{2} ln \space 2 + arctan \space 2\).

    34. \[\int_0^1 \int_1^2 \frac{y}{x + y^2} dy \space dx\]

    Exercise \(\PageIndex{35-38}\)

    In the following exercises, find the average value of the function over the given rectangles.

    35. \(f(x,y) = −x +2y\), \(R = [0,1] \times [0,1]\)

    Answer:

    \(\frac{1}{2}\).

    36. \(f(x,y) = x^4 + 2y^3\), \(R = [1,2] \times [2,3]\)

    37. \(f(x,y) = sinh \space x + sinh \space y\), \(R = [0,1] \times [0,2]\)

    Answer:

    \(\frac{1}{2}(2 \space cosh \space 1 + cosh \space 2 - 3)\).

    38. \(f(x,y) = arctan(xy)\), \(R = [0,1] \times [0,1]\)

    Exercise \(\PageIndex{39}\)

    39. Let f and g be two continuous functions such that \(0 \leq m_1 \leq f(x) \leq M_1\) for any \(x ∈ [a,b]\) and \(0 \leq m_2 \leq g(y) \leq M_2\) for any\( y ∈ [c,d]\). Show that the following inequality is true:

    \[m_1m_2(b-a)(c-d) \leq \int_a^b \int_c^d f(x) g(y) dy dx \leq M_1M_2 (b-a)(c-d).\]

    Exercise \(\PageIndex{40-43}\)

    In the following exercises, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.

    40. \(\frac{1}{e^2} \leq \iint_R e^{-x^2 - y^2} \space dA \leq 1\), where \(R = [0,1] \times [0,1]\)

    41. \(\frac{\pi^2}{144} \leq \iint_R sin \space x \space cos y \space dA \leq \frac{\pi^2}{48}\), where \(R = \left[ \frac{\pi}{6}, \frac{\pi}{3}\right] \times \left[ \frac{\pi}{6}, \frac{\pi}{3}\right]\)

    42. \(0 \leq \iint_R e^{-y}\space cos x \space dA \leq \frac{\pi}{2}\), where \(R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\)

    43. \(0 \leq \iint_R (ln \space x)(ln \space y) dA \leq (e - 1)^2\), where \(R = [1, e] \times [1, e] \)

    Exercise \(\PageIndex{44}\)

    44. Let f and g be two continuous functions such that \(0 \leq m_1 \leq f(x) \leq M_1\) for any \(x ∈ [a,b]\) and \(0 \leq m_2 \leq g(y) \leq M_2\) for any \(y ∈ [c,d]\). Show that the following inequality is true:

    \((m_1 + m_2) (b - a)(c - d) \leq \int_a^b \int_c^d |f(x) + g(y)| \space dy \space dx \leq (M_1 + M_2)(b - a)(c - d)\).

    Exercise \(\PageIndex{45-48}\)

    In the following exercises, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.

    45. \(\frac{2}{e} \leq \iint_R (e^{-x^2} + e^{-y^2}) dA \leq 2\), where \(R = [0,1] \times [0,1]\)

    46. \(\frac{\pi^2}{36}\iint_R (sin \space x + cos \space y)dA \leq \frac{\pi^2 \sqrt{3}}{36}\), where \(R = [\frac{\pi}{6}, \frac{\pi}{3}] \times [\frac{\pi}{6}, \frac{\pi}{3}]\)

    47. \(\frac{\pi}{2}e^{-\pi/2} \leq \iint_R (cos \space x + e^{-y})dA \leq \pi\), where \(R = [0, \frac{\pi}{2}] \times [0, \frac{\pi}{2}]\)

    48. \(\frac{1}{e} \leq \iint_R (e^{-y} - ln \space x) dA \leq 2\), where \(R = [0, 1] \times [0, 1]\)

    Exercise \(\PageIndex{49-50}\)

    In the following exercises, the function f is given in terms of double integrals.

    1. Determine the explicit form of the function f.
    2. Find the volume of the solid under the surface \(z = f(x,y)\) and above the region R.
    3. Find the average value of the function f on R.
    4. Use a computer algebra system (CAS) to plot \(z = f(x,y)\) and \(z = f_{ave}\) in the same system of coordinates.

    49. [T] \(f(x,y) = \int_0^y \int_0^x (xs + yt) ds \space dt\), where \((x,y) \in R = [0,1] \times [0,1]\)

    Answer:

    a. \(f(x,y) = \frac{1}{2} xy (x^2 + y^2)\); b. \(V = \int_0^1 \int_0^1 f(x,y) dx \space dy = \frac{1}{8}\); c. \(f_{ave} = \frac{1}{8}\);

    d.

    In xyz space, a plane is formed at z = 1/8, and there is another shape that starts at the origin, increases through the plane in a line roughly running from (1, 0.25, 0.125) to (0.25, 1, 0.125), and then rapidly increases to (1, 1, 1).

     

     50. [T] \(f(x,y) = \int_0^x \int_0^y [cos \space (s) + cos \space (t)] dt \space ds\), where \((x,y) \in R = [0,3] \times [0,3]\)

    Exercise \(\PageIndex{51-52}\)

    51. Show that if f and g are continuous on \([a,b]\) and \([c,d]\), respectively, then

    \(\int_a^b \int_c^d |f(x) + g(y)| dy \space dx = (d - c) \int_a^b f(x)dx\)

    \(+ \int_a^b \int_c^d g(y)dy \space dx = (b - a) \int_c^d g(y)dy + \int_c^d \int_a^b f(x) dx \space dy\).

    52. Show that \(\int_a^b \int_c^d yf(x) + xg(y) dy \space dx = \frac{1}{2} (d^2 - c^2) \left(\int_a^b f(x)dx\right) + \frac{1}{2} (b^2 - a^2) \left(\int_c^d g(y)dy\right)\).

     

    Exercise \(\PageIndex{53-54}\)

     53. [T] Consider the function \(f(x,y) = e^{-x^2-y^2}\), where \((x,y) \in R = [−1,1] \times [−1,1]\).

    1. Use the midpoint rule with \(m = n = 2,4,..., 10\) to estimate the double integral \(I = \iint_R e^{-x^2 - y^2} dA\). Round your answers to the nearest hundredths.
    2. For \(m = n = 2\), find the average value of f over the region R. Round your answer to the nearest hundredths.
    3. Use a CAS to graph in the same coordinate system the solid whose volume is given by \(\iint_R e^{-x^2-y^2} dA\) and the plane \(z = f_{ave}\).
    Answer:

    a. For \(m = n = 2\), \(I = 4e^{-0.5} \approx 2.43\) b. \(f_{ave} = e^{-0.5} \simeq 0.61\);

    c.

    In xyz space, a plane is formed at z = 0.61, and there is another shape with maximum roughly at (0, 0, 0.92), which decreases along all the sides to the points (plus or minus 1, plus or minus 1, 0.12).

    54. [T] Consider the function \(f(x,y) = sin \space (x^2) \space cos \space (y^2)\), where \((x,y \in R = [−1,1] \times [−1,1]\).

    1. Use the midpoint rule with \(m = n = 2,4,..., 10\) to estimate the double integral \(I = \iint_R sin \space (x^2) \space cos \space (y^2) \space dA\). Round your answers to the nearest hundredths.
    2. For \(m = n = 2\), find the average value of f over the region R. Round your answer to the nearest hundredths.
    3. Use a CAS to graph in the same coordinate system the solid whose volume is given by \(\iint_R sin \space (x^2) \space cos \space (y^2) \space dA\) and the plane \(z = f_{ave}\).

    Exercise \(\PageIndex{55-56}\)

    In the following exercises, the functions fnfn are given, where \(n \geq 1\) is a natural number.

    1. Find the volume of the solids \(S_n\) under the surfaces \(z = f_n(x,y)\) and above the region R.
    2. Determine the limit of the volumes of the solids \(S_n\) as n increases without bound.

    55. \(f(x,y) = x^n + y^n + xy, \space (x,y) \in R = [0,1] \times [0,1]\)

    Answer:

    a. \(\frac{2}{n + 1} + \frac{1}{4}\) b. \(\frac{1}{4}\)

    56. \(f(x,y) = \frac{1}{x^n} + \frac{1}{y^n}, \space (x,y) \in R = [1,2] \times [1,2]\)

    Exercise \(\PageIndex{57}\)

    57. Show that the average value of a function f on a rectangular region \(R = [a,b] \times [c,d]\) is \(f_{ave} \approx \frac{1}{mn} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*,y_{ij}^*)\),where \((x_{ij}^*,y_{ij}^*)\) are the sample points of the partition of R, where \(1 \leq i \leq m\) and \(1 \leq j \leq n\).

    Exercise \(\PageIndex{58}\)

    58. Use the midpoint rule with \(m = n\) to show that the average value of a function f on a rectangular region \(R = [a,b] \times [c,d]\) is approximated by

    \[f_{ave} \approx \frac{1}{n^2} \sum_{i,j =1}^n f \left(\frac{1}{2} (x_{i=1} + x_i), \space \frac{1}{2} (y_{j=1} + y_j)\right).\]

    Exercise \(\PageIndex{59}\)

    59. An isotherm map is a chart connecting points having the same temperature at a given time for a given period of time. Use the preceding exercise and apply the midpoint rule with \(m = n = 2\) to find the average temperature over the region given in the following figure.

    A contour map showing surface temperature in degrees Fahrenheit. Given the map, the midpoint rule would give rectangles with values 71, 72, 40, and 43.

    Answer

    \(56.5^{\circ}\) F; here \(f(x_1^*,y_1^*) = 71, \space f(x_2^*, y_1^*) = 72, \space f(x_2^*,y_1^*) = 40, \space f(x_2^*,y_2^*) = 43\), where \(x_i^*\) and \(y_j^*\) are the midpoints of the subintervals of the partitions of [a,b] and [c,d], respectively