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Chapter 13: Q 58. (page 1014)

Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
${鈭�}_{0}^{3} {鈭�}_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y$

Short Answer

Expert verified

The value of integral is $18$ .

Step by step solution

01

Given Information

Iterated integral is ${鈭�}_{0}^{3} {鈭�}_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y$

02

Calculation of Graph

The graph is as below:

03

Simplification

Changing the order of integral

${鈭�}_{0}^{3} {鈭�}_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y = {鈭�}_{0}^{3} {鈭�}_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d y d x$

Therefore

${鈭�}_{0}^{3} {鈭�}_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d y d x = {鈭�}_{0}^{3} [y]_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d x$

$= {鈭�}_{0}^{3} (9 - x^{2}) d x$

$鈬� {鈭�}_{0}^{3} {鈭�}_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d y d x = {[9 x - \frac{x^{3}}{3}]}_{0}^{3}$

$18$

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Most popular questions from this chapter

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

In the following lamina, all angles are right angles and the density is constant:

$How many summands are in {鈭�}_{j = 3}^{13} {鈭�}_{k = 5}^{20} \frac{j^{2}}{e^{jk}} ?$

Evaluate the iterated integral :

${鈭�}_{0}^{1} 鈥� {鈭�}_{2}^{4} 鈥� {鈭�}_{鈭� 1}^{5} 鈥� (x + y z^{2}) d x d y d z$

Evaluate the sums in Exercises $23 鈥� 28$ .
${鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{4} (3 j - 4 k)$

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