Chapter 13: Q. 26 (page 1055)

Evaluate the iterated integral :
$\int_{1}^{3} \int_{- 1}^{5} \int_{0}^{3} x y^{2} z d z d y d x$

Short Answer

Expert verified

$\frac{189}{2}$

Step by step solution

Step 1. Given information.

Given integral is :

$\int_{1}^{3} \int_{- 1}^{5} \int_{0}^{3} x y^{2} z d z d y d x$

We have to evaluate it.

Step 2. Evaluate.

Given $\int_{1}^{3} \int_{- 1}^{5} \int_{0}^{3} x y^{2} z d z d y d x$

$\begin{aligned} = \int_{1}^{3} \int_{- 1}^{5} (\int_{0}^{3} x y^{2} z d z) d y d x \\ = \int_{1}^{3} \int_{- 1}^{5} (x y^{2} \int_{0}^{3} z d z) d y d x \\ = \int_{1}^{3} \int_{- 1}^{5} (x y^{2} {(\frac{z^{2}}{2})}_{0}^{3}) d y d x \\ = \int_{1}^{3} \int_{- 1}^{5} (x y^{2} (\frac{9}{2})) d y d x \end{aligned}$

Step 3. Integrate with respect to y.

$\begin{aligned} = \int_{1}^{3} (\frac{9}{2} x \int_{- 1}^{5} y^{2} d y) d x \\ = \int_{1}^{3} (\frac{9}{2} x {(\frac{y^{3}}{3})}_{- 1}^{5} d x \\ = \int_{1}^{3} (\frac{9}{2} x (\frac{5^{3}}{3} - \frac{(- 1)^{3}}{3}) d x \\ = \int_{1}^{3} (\frac{9}{2} x (42)) d x \\ = \int_{1}^{2} 189 x d x \end{aligned}$

Integrate with respect to x.

$\begin{aligned} = 189 \int_{0}^{1} x d x \\ = 189 {(\frac{x^{2}}{2})}_{0}^{1} \\ = 189 (\frac{1}{2} - 0) \\ = \frac{189}{2} \end{aligned}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Evaluate the iterated integral :
$\int_{1}^{3} \int_{- 1}^{5} \int_{0}^{3} x y^{2} z d z d y d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to y.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Logic and Functions

Probability and Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Evaluate the iterated integral :∫13 ∫−15 ∫03 xy2zdzdydx

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

Applied Mathematics

Decision Maths

Logic and Functions

Probability and Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Evaluate the iterated integral :
$\int_{1}^{3} \int_{- 1}^{5} \int_{0}^{3} x y^{2} z d z d y d x$