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For each part below, calculate the average value of over the given interval. You may solve analytically or use your graphing calculator to evaluate.

  1. over

  2. over

Examine the Math Notes box about about Mean (average) Value. Notice that there are two strategies to compute Mean (average) Value.

Strategy 1: Compute the average area under the curve.
Strategy 2: Compute a slope of a secant line.

WARNING: These strategies can NOT be used interchangeably. The strategy you choose depends on the function you are given:
or .

And it depends on the question you are being asked:
Find average y-value? or Find average rate of change?

Average (Mean) Values

To calculate the mean (average) value of a finite set of items, add up the values of items and divide by the number of items.

Integrals help us add over a continuous interval. Therefore, for any continuous function :

mean value of over

First quadrant, bell curve labeled, f of x, left end point on the y axis, labeled, a, right end point labeled, b, dashed horizontal segment, about 1 fourth up from x axis to peak, labeled average, & shaded rectangle between, A & b, segment & x axis.

Since , we can also calculate the average value of any function using its antiderivative . Its average slope gives the average rate of change of , which is the same as the average value of

  mean rate of change of over 

First quadrant, 2 tick marks on x axis, first at the origin labeled, A, second almost to the right end, labeled b, Increasing curve labeled, capital F of x, starting at the origin, changing from concave up to concave down, in center of quadrant, ending at point corresponding to, b, almost at the top, with dashed segment labeled, m = average, from origin to end point of curve.