   Chapter 7.1, Problem 66E

Chapter
Section
Textbook Problem

# Calculate the average value of f ( x ) = x  sec 2 x on the interval [ 0 , π / 4 ] .

To determine

To evaluate: the average value of a function in the given interval.

Explanation

The average value of a function over some interval is the ratio of the sum of all function values in that range to the range of the interval.

Formula used:

The average value of a function over the interval [a,b] is given by the following integral:

avg=1baabf(x)dx

Given:

The function, f(x)=xsec2x and

the interval [0,π4]

Calculation:

Substitute the given function into the formula for average value with a and b as 0 and π4 respectively:

avg=1π400π4xsec2xdx=4π0π4xsec2xdx …… (1)

Solve the integral 0π4xsec2xdx using integration by parts. The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation. The formula for integration by parts for definite integral is given by

abf(x)g(x)dx=f(x)g(x)]ababg(x)f(x)dx

Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate

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