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Double Integrals Part 1 (Exercises)

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In exercises 1 and 2, 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 z=0.

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

Answer
27

2) f(x,y)=16x2+y2

In exercises 3 and 4, 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 sub-rectangles of the partition.

3) f(x,y)=sinxcosy, R=[0,π]×[0,π]

Answer
0

4) f(x,y)=cosx+cosy, R=[0,π]×[0,π2]

5) Use the midpoint rule with m=n=2 to estimate Rf(x,y)dA, where the values of the function f on R=[8,10]×[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]×[7,9] are given in the following table. Estimate the double integral Rf(x,y)dA by using a Riemann sum with m=n=2. Select the sample points to be the upper right corners of the sub-squares of R.

  y0=7 y1=8 y2=9
x0=0 10.22 10.21 9.85
x1=1 6.73 9.75 9.63
x2=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]×[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 ft3
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 sub-squares 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

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 to estimate the double integral Rf(x,y)dA, where R=[0.2,1]×[0,0.8].
  2. Estimate the average value of the function f on R.

    CNX_Calc_Figure_15_01_201.jpg

Answer
a. Answers will vary somewhat, but should be close to these: f(0.4,0.2)0.125, f(0.4,0.6)0.2, f(0.8,0.2)0.6, and f(0.8,0.6)0.3.
Since the area of each rectangle is ΔAi=(0.4)(0.4)=0.16 units2, Rf(x,y)dA=10.20.80f(x,y)dydxf(0.4,0.2)(0.16)+f(0.4,0.6)(0.16)+f(0.8,0.2)(0.16)+f(0.8,0.6)(0.16)=0.132
b. fave=10.20.80f(x,y)dydxarea of R10.20.80f(x,y)dydx0.640.206

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 Rf(x,y)dA, where R=[0.1,0.5]×[0.1,0.5].
  2. Estimate the average value of the function f on R.

    CNX_Calc_Figure_15_01_202.jpg

11) The solid lying under the surface z=4y2 and above the rectangular regionR=[0,2]×[0,2] is illustrated in the following graph. Evaluate the double integral Rf(x,y), where f(x,y)=4y2 by finding the volume of the corresponding solid.

Answer
2π

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

CNX_Calc_Figure_15_01_204.jpg

In the exercises 13 - 20, calculate the integrals by reversing the order of integration.

13) 11(22(2x+3y+5)dx) dy

Answer
40

14) 20(10(x+2ey+3)dx) dy

15) 271(21(3x+3y)dy) dx

Answer
812+3932

16) 161(81(4x+23y)dy) dx

17) ln3ln2(10ex+ydy) dx

Answer
e1

18) 20(103x+ydy) dx

19) 61(92yx2dy) dx

Answer
151029

20) 91(24xy2dy)dx

In exercises 21 - 34, evaluate the iterated integrals by choosing the order of integration.

21) π0π/20sin(2x)cos(3y)dx dy

Answer
0

22) π/8π/12π/3π/4[cotx+tan(2y)]dx dy

23) e1e1[1xsin(lnx)+1ycos(lny)]dx dy

Answer
(e1)(1+sin1cos1)

24) e1e1sin(lnx)cos(lny)xydx dy

25) 2121(lnyx+x2y+1)dy dx

Answer
34ln(53)+2(ln2)2ln2

26) e121x2ln(x)dy dx

27) 3121y arctan(1x)dy dx

Answer
18[(233)π+6 ln2]

28) 101/20(arcsinx+arcsiny)dy dx

29) 1021xex+4ydy dx

Answer
14e4(e41)

30) 2110xexydy dx

31) e1e1(lnyy+lnxx)dy dx

Answer
4(e1)(2e)

32) e1e1(x lnyy+y lnxx)dy dx

33) 1021(xx2+y2)dy dx

Answer
π4+ln(54)12ln2+arctan2

34) 1021yx+y2dy dx

In exercises 35 - 38, find the average value of the function over the given rectangles.

35)f(x,y)=x+2y, R=[0,1]×[0,1]

Answer
12

36) f(x,y)=x4+2y3, R=[1,2]×[2,3]

37) f(x,y)=sinhx+sinhy, R=[0,1]×[0,2]

Answer
12(2 cosh1+cosh23).

38) f(x,y)=arctan(xy), R=[0,1]×[0,1]

39) Let f and g be two continuous functions such that 0m1f(x)M1 for any x[a,b] and 0m2g(y)M2 for anyy[c,d]. Show that the following inequality is true:

m1m2(ba)(cd)badcf(x)g(y)dydxM1M2(ba)(cd).

In exercises 40 - 43, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.

40) 1e2Rex2y2 dA1, where R=[0,1]×[0,1]

41) π2144Rsinxcosy dAπ248, where R=[π6,π3]×[π6,π3]

42) 0Rey cosx dAπ2, where R=[0,π2]×[0,π2]

43) 0R(lnx)(lny)dA(e1)2, where R=[1,e]×[1,e]

44) Let f and g be two continuous functions such that 0m1f(x)M1 for any x[a,b] and 0m2g(y)M2 for any y[c,d]. Show that the following inequality is true:

(m1+m2)(ba)(cd)badc|f(x)+g(y)| dy dx(M1+M2)(ba)(cd)

In exercises 45 - 48, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.

45) 2eR(ex2+ey2)dA2, where R=[0,1]×[0,1]

46) π236R(sinx+cosy)dAπ2336, where R=[π6,π3]×[π6,π3]

47) π2eπ/2R(cosx+ey)dAπ, where R=[0,π2]×[0,π2]

48) 1eR(eylnx)dA2, where R=[0,1]×[0,1]

In exercises 49 - 50, 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.

CNX_Calc_Figure_15_01_205.jpg

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

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

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

\displaystyle + \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 \displaystyle \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).

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.

CNX_Calc_Figure_15_01_207.jpg

54) [T] Consider the function f(x,y) = \sin (x^2) \space \cos (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 (x^2) \cos (y^2) \space dA. Round your answers to the nearest hundredths.
  2. For m = n = 2, find the average value of fover 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(x^2) \cos(y^2) \space dA and the plane z = f_{ave}.

In exercises 55 - 56, the functions f_n 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]

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.

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). \nonumber

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.

CNX_Calc_Figure_15_01_209.jpg

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.

 


Double Integrals Part 1 (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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