### Problem Statement:**Find the mass of the lamina that occupies the region $ D $ and has the given density function $ \rho $.**Region $ D $ is defined as the triangular region enclosed by the lines $ x = 0 $, $ y = x $, and $ 2x + y = 6 $. The density function is given by:\[ \rho(x, y) = 8x^2 \]### Solution:To determine the mass $ m $ of the lamina, we need to integrate the density function over the region $ D $.### Steps to Calculate the Mass:1. **Define the Region $ D $:** The region $ D $ is the triangular region enclosed by the lines $ x = 0 $, $ y = x $, and $ 2x + y = 6 $. 2. **Set up the Integral:** To find the total mass $ m $, we use the double integral of the density function over the region $ D $: \[ m = \iint_D \rho(x, y) \, dA \]### Explanation of the Integral:The bounds for the integral need to be determined based on the lines that form the triangular region:- For $ x = 0 $ to line $ 2x + y = 6 $: \[ y = 6 - 2x \]- For each $ x $, $ y $ ranges from $ x $ to $ 6 - 2x $.Therefore, the integral for mass $ m $ can be set up as:\[ m = \int_{0}^{3} \int_{x}^{6 - 2x} 8x^2 \, dy \, dx \]Where:- The outer integral ($ dx $) ranges from 0 to 3 because $ x $ ranges from 0 (where $ x = 0 $ line) to 3 (where $ x = 3 $ line intersects $ 2x + y = 6 $).- The inner integral ($ dy $) ranges from $ y = x $ to $ y = 6 - 2x $.By solving this integral, we obtain the mass $ m $ of the lamina. \[m = \int_{0

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Find the mass of the lamina that occupies the region D and has the given density function p. D is the triangular region enclosed by the lines x = 0, y = x, and 2x + y = 6; p(x, y) = 8x² %3D m =

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Problem Statement:

**Find the mass of the lamina that occupies the region $ D $ and has the given density function $ \rho $.**

Region $ D $ is defined as the triangular region enclosed by the lines $ x = 0 $, $ y = x $, and $ 2x + y = 6 $. The density function is given by:

\[ \rho(x, y) = 8x^2 \]

### Solution:

To determine the mass $ m $ of the lamina, we need to integrate the density function over the region $ D $.

### Steps to Calculate the Mass:

1. **Define the Region $ D $:**
The region $ D $ is the triangular region enclosed by the lines $ x = 0 $, $ y = x $, and $ 2x + y = 6 $.

2. **Set up the Integral:**
To find the total mass $ m $, we use the double integral of the density function over the region $ D $:

\[ m = \iint_D \rho(x, y) \, dA \]

### Explanation of the Integral:

The bounds for the integral need to be determined based on the lines that form the triangular region:

- For $ x = 0 $ to line $ 2x + y = 6 $:
\[
y = 6 - 2x
\]
- For each $ x $, $ y $ ranges from $ x $ to $ 6 - 2x $.

Therefore, the integral for mass $ m $ can be set up as:

\[ m = \int_{0}^{3} \int_{x}^{6 - 2x} 8x^2 \, dy \, dx \]

Where:
- The outer integral ($ dx $) ranges from 0 to 3 because $ x $ ranges from 0 (where $ x = 0 $ line) to 3 (where $ x = 3 $ line intersects $ 2x + y = 6 $).

- The inner integral ($ dy $) ranges from $ y = x $ to $ y = 6 - 2x $.

By solving this integral, we obtain the mass $ m $ of the lamina.

\[
m = \int_{0

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