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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding the Area Bounded by Two Graphs In Exercises 15-30, sketch the region bounded by the graphs of the functions and find the area of the region. See Examples 1, 2, 3, and 4.

y = 8 x 2 , y = x 2

To determine

To graph: The region bounded by the graphs of y=8x2 and y=x2, and also compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of y=8x2 and y=x2.

Graph:

Consider the provided functions,

y=8x2 and y=x2

The function y=8x2 is a quadratic function.

And the term x2 is negative, so its graph will be a parabola with opening downwards. Compute its x-intercepts as follows,

8x2=0x2=8x=±22

The x-intercepts are x=±22.

Now the second function y=x2.

It is a quadratic function. The term x2 is positive, so its graph will be a parabola with opening upwards.

Now compute its x-intercepts as,

x2=0x=0

The x-intercepts are x=0.

Compute the points of intersection of two graphs by setting the functions equal to each other and solving for x,

8x2=x22x2=8x2=4x=2, 2

Substitute x=2 in the function y=x2 and compute the first intersection point,

y=22=4

Substitute x=2 in the function y=x2 and compute the second intersection point,

y=(2)2=4

So, the graphs of given functions intersect at the points (2,4) and (2,4). These points give interval of integration [2,2].

Sketch the graph of two functions as follows:

Formula used:

Area of a region bounded by two graphs is calculated using the following formula,

If f and g are continuous on [a,b] and g(x)f(x) for all x in [a,b], then the area of the region bounded by the graphs of f, g, x=a and x=b is given by,

A=ab[f(x)g(x)]dx

The integral of xndx=xn+1n+1+C

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