Chapter 14, Problem 56RE

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# VolumeIn Exercises 55 and 56, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z = 8 − x − y ,     z = 0 ,   y = x ,     y = 3 ,     x = 0

To determine

To calculate: The volume of z=8xy,z=0,y=x,y=3,x=0 for the solid bounded by graphs

Explanation

Given:

The solid is bounded by graphs z=8âˆ’xâˆ’y,z=0,y=x,y=3,x=0.

Formula used:

Integration formula:

âˆ«abxndx=[xn+1n+1]ab

Volume of a solid:

V=âˆ«y1y2âˆ«x1x2âˆ«z1z2dzdxdy

Calculation:

We consider the solid bounded by graph,

Apply the formula for volume of solid,

V=âˆ«y1y2âˆ«x1x2âˆ«z1z2dzdxdy

As can be seen from above formula

x1=0,x2=yy1=0,y2=3z1=0,z2=8âˆ’xâˆ’y

âˆ´,

V=âˆ«y1y2âˆ«x1x2âˆ«z1z2dzdxdy=âˆ«03âˆ«0yâˆ«08âˆ’xâˆ’ydzdxdy

Assume remaining variable constant and integrate the provided expression with respect to z

âˆ«03âˆ«0yâˆ«08âˆ’xâˆ’ydzdxd<

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