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Chapter 13: Q. 32 (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.
${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{4} {鈭�}_{r}^{4} r d z d r d 胃$

Short Answer

Expert verified

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

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�}_{r}^{4} r d z d r d 胃

02

Evaluate the integral

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

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Most popular questions from this chapter

Evaluate the iterated integral :

${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� {鈭�}_{0}^{\sin^{鈭� 1} 鈦� x} 鈥� (x + y) d z d y d x$

Let $惟$ be a lamina in the xy-plane. Suppose $惟$ is composed of n non-overlapping lamin忙 role="math" localid="1650321722341" $惟_{1}, 惟_{2}, 鈥�, 惟_{n} .$ Show that if the masses of these lamin忙 are $m_{1}, m_{2}, 鈥�, m_{n}$ and the centers of masses are $({\overset{炉}{x}}_{1}, {\overset{炉}{y}}_{1}), ({\overset{炉}{x}}_{2}, {\overset{炉}{y}}_{2}), 鈥�, ({\overset{炉}{x}}_{n}, {\overset{炉}{y}}_{n}),$ then the center of mass of $惟$ is $(\overset{炉}{x}, \overset{炉}{y}),$ where

$\overset{炉}{x} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{x}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} and \overset{炉}{y} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{y}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} .$

Discuss the similarities and differences between the definition of the double integral found in Section $13.1$ and the definition of the triple integral found in this section.

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

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

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