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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 5.5, Problem 18E
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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=8−x2 and y=x2, and also compute the area of the region.

Explanation

Given Information:

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

Graph:

Consider the provided functions,

y=8−x2 and y=x2

The function y=8−x2 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,

8−x2=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,

8−x2=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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All of us have a tendency to categorize people into stereotypic groups. If you look around at the students on c...

Research Methods for the Behavioral Sciences (MindTap Course List)

Find an equation of the parabola in standard form with vertex at the origin and focus at (3,0).

College Algebra (MindTap Course List)

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