   Chapter 5.5, Problem 12E

Chapter
Section
Textbook Problem

# (a) Find the average value of f on the given interval.(b) Find c such that f a v e = f ( c ) .(c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f. f ( x ) = x 4 − x 2 ,     [ 0 , 2 ]

To determine

(a)

To find:

The average value of f on the given interval.

Explanation

1) Concept:

Use the formula for the average value of f on the interval [a, b].

2) Formula:

fave=1b-aabf(x)dx

3) Given:

fx=x4-x2,     [0, 2]

4) Calculation:

Here, fx=x4-x2,  a=0 and  b=2

Substituting all the above in the formula,

fave=12-002x4-x2dx

Simplify.

fave=1202x4-x2dx

Here, we need to use the substitution method.

Let  u=4-x2 so du=-2xdx that is xdx=-du2

Since this is a definite integral, the limits also change with respect to the substitution.

When x=0, u=4 and when x=2, u=0

Substituting all in the above integration,

fave=1202x4-x2dx

=

To determine

(b)

To find:

The value of c such that fave=f(c)

To determine

(c)

To sketch:

The graph of f and rectangle whose area is same the area under the graph of f

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