# Question Video: Finding the Average Value of a Quadratic Function on a Given Interval Mathematics • Higher Education

Determine the average value of π(π₯) = 3π₯Β² β 2π₯ on the interval [β3, 5].

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### Video Transcript

Determine the average value of π of π₯ equals three π₯ squared minus two π₯ on the closed interval negative three to five.

Remember, the formula for the average value of the function π over some closed interval π to π is one over π minus π times the integral evaluated between π and π of π of π₯ with respect to π₯. In this case, we see that our π of π₯ is equal to three π₯ squared minus two π₯. Our closed interval is from negative three to five. So we let π be equal to negative three and π be equal to five. One over π minus π becomes one over five minus negative three. And this is multiplied by the integral evaluated between negative three and five of three π₯ squared minus two π₯ with respect to π₯. Five minus negative three is eight. So we need to evaluate this definite integral.

Here, we recall that the indefinite integral of a general polynomial term ππ₯ to the power of π is ππ₯ to the power of π plus one over π plus one plus π, where π and π are constants and π is not equal to negative one. We also recall that we can integrate the sum of polynomial terms by integrating each term individually. And we see that the integral of three π₯ squared is three π₯ cubed over three. And the integral of negative two π₯ is negative two π₯ squared over two. And weβre going to evaluate this between negative three and five.

Now this simplifies somewhat to π₯ cubed minus π₯ squared. So letβs substitute our limits in. Weβre looking to find an eighth of five cubed minus five squared minus negative three cubed minus negative three squared. Thatβs an eighth of 100 minus negative 36, which is equal to 17. So the average value of our function three π₯ squared minus two π₯ on the closed interval negative three to five is 17.