Q. What is the average value of x3−9x on the interval −1≤x≤3 ?
Set up integral: To find the average value of the function x3−9x on the interval from −1 to 3, we need to integrate the function over the interval and then divide by the length of the interval.
Find antiderivative: First, we set up the integral of the function x3−9x from −1 to 3. Integral from −1 to 3 of (x3−9x)dx
Evaluate antiderivative: Next, we find the antiderivative of x3−9x. The antiderivative of x3 is (1/4)x4, and the antiderivative of −9x is (−9/2)x2. So, the antiderivative of x3−9x is (1/4)x4−(9/2)x2.
Calculate interval length: We evaluate the antiderivative from −1 to 3. [41x4−29x2] evaluated from −1 to 3 = [41(3)4−29(3)2]−[41(−1)4−29(−1)2] = [41(81)−29(9)]−[41(1)−29(1)] = [481−281]−[41−29] = [20.25−40.5]−[0.25−4.5] = −20.25+4.25 = 30
Divide for average value: Now, we calculate the length of the interval from −1 to 3.Length of the interval = 3−(−1)=3+1=4
Divide for average value: Now, we calculate the length of the interval from −1 to 3. Length of the interval = 3−(−1)=3+1=4 Finally, we divide the result of the integral by the length of the interval to find the average value. Average value = Integral result / Length of the interval Average value = −16/4 Average value = −4
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