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c) Average value of
f(x)=x^(3) on
[5,13] :
Note: You can earn partial credit on this problem.
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c) Average value of $f(x)=x^{3}$ on $[5,13]$ : $\newline$ Note: You can earn partial credit on this problem. $\newline$ Preview My Answers $\newline$ Submit Answers

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Power rule with rational exponents

Full solution

Q. c) Average value of

f(x)=x^{3}

on

[5,13]

:

\newline

Note: You can earn partial credit on this problem.

\newline

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\newline

Submit Answers

Identify Interval: To find the average value of a function on an interval $[a, b]$ , we use the formula: Average value = $\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx$ .
Calculate Integral: First, let's identify $a$ and $b$ for our interval $[5,13]$ . So, $a = 5$ and $b = 13$ .
Find Antiderivative: Now, we need to calculate the definite integral of $f(x) = x^3$ from $5$ to $13$ .
Evaluate Limits: The antiderivative of $x^3$ is $(1/4)x^4$ . So, we will evaluate $(1/4)x^4$ from $5$ to $13$ .
Substitute Values: Plugging in the limits, we get $(\frac{1}{4})(13^4) - (\frac{1}{4})(5^4)$ .
Simplify Expression: Calculating the powers, $13^4 = 28561$ and $5^4 = 625$ .
Calculate Powers: Now, we substitute these values into our expression: $(\frac{1}{4})(28561) - (\frac{1}{4})(625)$ .
Subtract Values: This simplifies to $7140.25 - 156.25$ .
Divide by Interval: Subtracting these values gives us $6984$ .
Find Average Value: Finally, we divide by $(b-a)$ , which is $(13-5) = 8$ , to find the average value.
Find Average Value: Finally, we divide by $(b-a)$ , which is $(13-5) = 8$ , to find the average value.So, the average value is $6984 / 8$ .

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