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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Section
BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

The figure shows a curve C with the property that, for every point P on the middle curve y = 2 x 2 , the areas A and B are equal. Find an equation for C.

Chapter 5.P, Problem 9P, The figure shows a curve C with the property that, for every point P on the middle curve y=2x2, the

To determine

To find:

Equation for C

Explanation

1) Concept:

Construct integrals for area A and B, and then find the equation of curve C.

2) Calculation:

As point P lies on the curve y=2x2, with x=a, the y  co-ordinate of point P is 2a2.

So, let coordinates of point P be (a, 2a2).

To calculate the area A:

It is the area between curve y=2x2 and y=x2, between x=0 and x=a. So

A=0a2x2-x2dx

A=0ax2dx

A=x330a

A=a33

To calculate area B: It is the area between curves C and y=2x2 where y lies between 0 and a2.

It lies in the first quadrant.

Solve y=2x2 for x.

So,

x= fy=y2

Consider the equation for curve C as x=C(y).

So the area between curves is given by,

B=02a2y2-C(y)dy

B=43y23202a2-02a2Cydy

B=43a3-02

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