187k views
0 votes
The function f(x)=x+x⁴ is given. Determine the average value of this function over the interval [−5,4].

User Ukonn Ra
by
8.0k points

1 Answer

2 votes

Final answer:

To find the average value of the function f(x) = x + x⁴ on the interval [−5,4], integrate the function over this range and divide the result by the interval length, which is 9.

Step-by-step explanation:

To find the average value of the function f(x) = x + x⁴ over the interval [−5,4], you use the formula for the average value of a function over an interval [a, b], which is:

\[\frac{1}{b - a}\int_{a}^{b} f(x) dx\]

First, integrate the function from −5 to 4:

\[\int_{-5}^{4} (x + x⁴) dx\] = \left[ \frac{x^2}{2} + \frac{x^5}{5} \right]_{-5}^{4}

Next, evaluate the integral at the limits of integration:

\[= \left( \frac{4^2}{2} + \frac{4^5}{5} \right) - \left( \frac{(-5)^2}{2} + \frac{(-5)^5}{5} \right)\]

Now, calculate this to find the integral's value, and divide the result by the length of the interval, which is 4 - (-5) = 9.

Finally, you get the average value by applying the formula:

\[\frac{1}{9}(\text{Value of Integral})\]

Remember to perform the calculations step by step to get the final value for the average.

User Seraf
by
7.8k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories