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Chapter 13: Q. 33 (page 1083)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{4} {鈭�}_{0}^{4 - r} r d z d r d 胃$

Short Answer

Expert verified

${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{4} {鈭�}_{0}^{4 - r} r d z d r d 胃 = \frac{64}{3} 蟺$

Step by step solution

01

Given

We have to find the volume by using the triple integral.

{鈭�}_{0}^{2 蟺} {鈭�}_{0}^{4} {鈭�}_{0}^{4 - r} r d z d r d 胃

02

Evaluate the integral

$V = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{4} [{鈭�}_{0}^{4 - r} r d z] d r d 胃 V = {鈭�}_{0}^{2 蟺} [{鈭�}_{0}^{4} r (4 - r) d r] d 胃 V = {鈭�}_{0}^{2 蟺} {[4 \frac{r^{2}}{2} - \frac{r^{3}}{3}]}_{0}^{4} d 胃 V = {鈭�}_{0}^{2 蟺} [32 - \frac{64}{3}] d 胃 V = \frac{32}{3} {鈭�}_{0}^{2 蟺} d 胃 V = \frac{32}{3} (2 蟺 - 0) V = \frac{64}{3} 蟺 cubic units$

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Most popular questions from this chapter

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$鈭� {鈭�}_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{蟺}{4} 鈮� x 鈮� \frac{蟺}{2} a n d \frac{蟺}{2} 鈮� y 鈮� 蟺\}$

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.

localid="1650380493598" $鈭� {鈭�}_{R} \sin (2 x + y) d A,$

wherelocalid="1650380496793" $R = \{(x, y) | 0 鈮� x 鈮� \frac{蟺}{2} a n d 0 鈮� y 鈮� \frac{蟺}{2}\}$

How many summands are in ${鈭�}_{i = i_{0}}^{l} {鈭�}_{j = j_{0}}^{m} {鈭�}_{k = k_{0}}^{n} k^{i} j^{k}$ ?

Use the lamina from Exercise 64, but assume that the density is proportional to the distance from the x-axis.

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{0}^{2} {鈭�}_{0}^{1 - 3 x / 2} {鈭�}_{2 x + (4 / 3) y - 4}^{0} f (x, y, z) d z d y d x$

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