Determine the average value of 𝑓 of 𝑥 equals three 𝑥 squared minus two 𝑥 on the closed interval negative three to five.
Remember, the formula for the average value of the function 𝑓 over some closed interval 𝑎 to 𝑏 is one over 𝑏 minus 𝑎 times the integral evaluated between 𝑎 and 𝑏 of 𝑓 of 𝑥 with respect to 𝑥. In this case, we see that our 𝑓 of 𝑥 is equal to three 𝑥 squared minus two 𝑥. Our closed interval is from negative three to five. So we let 𝑎 be equal to negative three and 𝑏 be equal to five. One over 𝑏 minus 𝑎 becomes one over five minus negative three. And this is multiplied by the integral evaluated between negative three and five of three 𝑥 squared minus two 𝑥 with respect to 𝑥. 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 𝑎𝑥 to the power of 𝑛 is 𝑎𝑥 to the power of 𝑛 plus one over 𝑛 plus one plus 𝑐, where 𝑎 and 𝑐 are constants and 𝑛 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 𝑥 squared is three 𝑥 cubed over three. And the integral of negative two 𝑥 is negative two 𝑥 squared over two. And we’re going to evaluate this between negative three and five.
Now this simplifies somewhat to 𝑥 cubed minus 𝑥 squared. So let’s substitute our limits in. We’re looking to find an eighth of five cubed minus five squared minus negative three cubed minus negative three squared. That’s an eighth of 100 minus negative 36, which is equal to 17. So the average value of our function three 𝑥 squared minus two 𝑥 on the closed interval negative three to five is 17.