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Chapter 5.4, Problem 49E
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### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

#### Solutions

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### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# The area of the region that lies to the right of the y-axis and to the left of the parabola x = 2y − y2 (the shaded region in the figure) is given by the integral ∫ 0 2 ( 2 y − y 2 )   d y . (Turn your head clockwise and think of the region as lying below the curve x = 2y − y2 from y = 0 to y = 2.) Find the area of the region.

To determine

To find: The area of the shaded region.

Explanation

Given information:

The shaded region is bounded by a curve with x=2yâˆ’y2.

The region lies between y=0 and y=2.

Find the area of the shaded region (A) as shown below:

A=âˆ«abxdy (1)

Here, the curve function is x, the lower limit is a, and the upper limit is b.

Substitute 2yâˆ’y2 for x, 2 for b, and 0 for a in Equation (1).

A=âˆ«02(2yâˆ’y2)dy=[2

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th