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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 , y = x 2 , y = 0 , x = 1 , x = 4

To determine

To graph: The region bounded by the graphs of y=8x, y=x2, y=0, x=1 and x=4, and also compute the graph of the region.

Explanation

Given Information:

The region bounded by the graphs of y=8x, y=x2, y=0, x=1 and x=4.

Graph:

Consider the following equations that give the required region,

y=8x, y=x2, y=0, x=1 and x=4

The first function y=8x is a reciprocal function. The graph will lie in first and third quadrant. As x increases in both directions, y will decrease and vice-versa.

There will be no x-intercept and no y-intercept.

Consider the second function y=x2

It is a quadratic function. So, its graph will be a parabola opening upwards. The x-intercept will be (0,0).

Consider the third function y=0

It is the equation of the x-axis.

Compute the intersection point of the graphs of y=8x and y=x2 as follows:

x2=8xx3=8x3=23x=2

Substitute x=2 in the equation y=8x and compute the intersection point,

y=82=4

So two graphs intersect at point (2,4).

Sketch the graph of the region 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

Power and log rule for integrals are,

xndx=xn+1n+1+C1xdx=ln|x|+C

Calculation:

From the above, the required area consists of two regions, left region and right region

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