Question

In Exercises \(31 - 36 ,\) find the average value of the function on the interval, using antiderivatives to compute the integral. $$y = \sec ^ { 2 } x , \quad \left[ 0 , \frac { \pi } { 4 } \right]$$

Short answer

The average value of the function on the given interval is \( \frac{4}{\pi} \).

Step-by-step solution

Identifying the Function and Interval

Identify the function and the interval given in the exercise. Here, the function is \(y = \sec^2x\) and the interval is \([0, \frac{\pi}{4}]\).

Computing the Antiderivative

Find the antiderivative of the function. The antiderivative of \( \sec^2x \) is \( \tan x \).

Applying the Fundamental Theorem of Calculus

Apply the Fundamental Theorem of Calculus to evaluate the integral over the interval. This requires substituting the endpoints of the interval into the antiderivative function and subtracting: \( \tan(\frac{\pi}{4}) - \tan(0) \) which simplifies to \( 1 - 0 = 1 \).

Finding the Average Value

Find the average value of the function on the interval by dividing the definite integral by the width of the interval. The width of the interval is \( \frac{\pi}{4} - 0 = \frac{\pi}{4} \). So the average value is \( \frac{1}{\frac{\pi}{4}} = \frac{4}{\pi} \).