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Chapter 13: Q. 30 (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.
${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{2} {鈭�}_{0}^{5} r d z d r d 胃$

Short Answer

Expert verified

${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{2} {鈭�}_{0}^{5} r d z d r d 胃 = 20 蟺$

Step by step solution

01

Given

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

{鈭�}_{0}^{2 蟺} {鈭�}_{0}^{2} {鈭�}_{0}^{5} r d z d r d 胃

02

Evaluate the integral

$V = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{2} [{鈭�}_{0}^{5} r d z] d r d 胃 V = {鈭�}_{0}^{2 蟺} [{鈭�}_{0}^{2} 5 r d r] d 胃 V = 5 {鈭�}_{0}^{2 蟺} {[\frac{r^{2}}{2}]}_{0}^{2} d 胃 V = 5 脳 2 {鈭�}_{0}^{2 蟺} d 胃 V = 10 (2 蟺 - 0) V = 20 蟺 cubic units$

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Most popular questions from this chapter

Explain why ${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} i j^{2} k^{3} = {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} {鈭�}_{i = 1}^{l} i j^{2} k^{3}$ .

Use the results of Exercises 59 and 60 to find the centers of masses of the lamin忙 in Exercises 61鈥�67.

Use the lamina from Exercise 61, but assume that the density is proportional to the distance from the x-axis.

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

$鈭� {鈭�}_{R} e^{x y} d A w h e r e R = \{(x, y) I 1 鈮� x 鈮� 2 a n d 1 鈮� y 鈮� 3\}$

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

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - z^{2}}}^{\sqrt{9 - z^{2}}} {鈭�}_{0}^{3 - \sqrt{x^{2} + z^{2}}} f (x, y, z) d y d x d z$

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral ${鈭�}_{a}^{b} {鈭�}_{c}^{d} f (x, y) d y d x$ .

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