Q. 28 Evaluate the iterated integral :... [FREE SOLUTION]

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Chapter 13: Q. 28 (page 1055)

Evaluate the iterated integral :
${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� {鈭�}_{0}^{6 y} 鈥� \frac{x}{y} d z d x d y$

Short Answer

Expert verified

$9$

Step by step solution

Step 1. Given information.

Given integral is :

${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� {鈭�}_{0}^{6 y} 鈥� \frac{x}{y} d z d x d y$

We have to evaluate it.

Step 2. Evaluate.

Given ${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� {鈭�}_{0}^{6 y} 鈥� \frac{x}{y} d z d x d y$

$\begin{aligned} = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� [({鈭�}_{0}^{6 y} 鈥� \frac{x}{y} d z) d x d y] \\ = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� [(\frac{x}{y} {鈭�}_{0}^{6 y} 鈥� d z) d x d y] \\ = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� [\frac{x}{y} (z)_{0}^{6 y}] d x d y \\ = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� [\frac{x}{y} (6 y)] d x d y \\ = {鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� (6 x) d x d y \end{aligned}$

Step 3. Integrate with respect to z.

Integrate with respect to z

$= {鈭�}_{0}^{1} 鈥� [({鈭�}_{0}^{3 鈭� 3 y} 鈥� 6 x d x) d y] = {鈭�}_{0}^{1} 鈥� (6 {(\frac{x^{2}}{2})}_{0}^{3 鈭� 3 y}) d y = {鈭�}_{0}^{1} 鈥� {(3 x^{2})}_{0}^{3 鈭� 3 y} d y \begin{aligned} = {鈭�}_{0}^{1} 鈥� 3 (3 鈭� 3 y)^{2} d y \\ = {鈭�}_{0}^{1} 鈥� 3 (9 鈭� 18 y + 9 y^{2}) d y \\ = 27 {鈭�}_{0}^{1} 鈥� d y 鈭� 54 {鈭�}_{0}^{1} 鈥� y d y + 27 {鈭�}_{0}^{1} 鈥� y^{2} d y \\ = 27 (y)_{0}^{1} 鈭� 54 {(\frac{y^{2}}{2})}_{0}^{1} + 27 {(\frac{y^{3}}{3})}_{0}^{1} \\ = 27 鈭� 27 + 9 \\ = 9 \end{aligned}$

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91影视

Evaluate the iterated integral :
${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� {鈭�}_{0}^{6 y} 鈥� \frac{x}{y} d z d x d y$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to z.

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Evaluate the iterated integral :鈭�01鈥�鈭�03鈭�3y鈥�鈭�06y鈥�xydzdxdy

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to z.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Applied Mathematics

Calculus

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

Evaluate the iterated integral :
${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3 鈭� 3 y} 鈥� {鈭�}_{0}^{6 y} 鈥� \frac{x}{y} d z d x d y$