Q. 30 Evaluate the iterated integral :... [FREE SOLUTION]

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Chapter 13: Q. 30 (page 1055)

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

Short Answer

Expert verified

$\frac{21 蟺}{8} 鈭� \frac{9}{2}$

Step by step solution

Step 1. Given information.

Given integral is :

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

We have to evaluate it.

Step 2. Evaluate.

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

$\begin{aligned} = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� [{鈭�}_{0}^{\sin^{鈭� 1} 鈦� x} 鈥� (x + y) d z] d y d x \\ = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� [(x + y) {鈭�}_{0}^{\sin^{鈭� 1} 鈦� x} 鈥� d z] d y d x \\ = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� [(x + y) [z]_{0}^{\sin^{鈭� 1} 鈦� x}] d y d x \\ = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� [(x + y) [\sin^{鈭� 1} 鈦� x]] d y d x \end{aligned}$

Step 3. Integrate with respect to y.

Integrate with respect to y

$\begin{aligned} = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� [(x + y) [\sin^{鈭� 1} 鈦� x]] d y d x \\ = {鈭�}_{0}^{1} 鈥� [x \sin^{鈭� 1} 鈦� x {鈭�}_{0}^{3} 鈥� d y + \sin^{鈭� 1} 鈦� x {鈭�}_{0}^{3} 鈥� y d y] d x \\ = {鈭�}_{0}^{1} 鈥� [x \sin^{鈭� 1} 鈦� x [y]_{0}^{3} + \sin^{鈭� 1} 鈦� x {[\frac{y^{2}}{2}]}_{0}^{3}] d x \\ = {鈭�}_{0}^{1} 鈥� [x \sin^{鈭� 1} 鈦� x [3] + \sin^{鈭� 1} 鈦� x [\frac{3^{2}}{2}]] d x \\ = 3 {鈭�}_{0}^{1} 鈥� x \sin^{鈭� 1} 鈦� x d x + \frac{9}{2} {鈭�}_{0}^{1} 鈥� \sin^{鈭� 1} 鈦� x d x \end{aligned}$

Use integration by parts :

$\begin{aligned} = 3 [\frac{蟺}{8}] + \frac{9}{2} [\frac{蟺}{2} 鈭� 1] \\ = [\frac{3 蟺}{8} + \frac{9 蟺}{4} 鈭� \frac{9}{2}] \\ = \frac{21 蟺}{8} 鈭� \frac{9}{2} \end{aligned}$

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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to y.

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Evaluate the iterated integral :鈭�01鈥�鈭�03鈥�鈭�0sin鈭�1鈦�x鈥�(x+y)dzdydx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to y.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Decision Maths

Calculus

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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