Video: Finding the Definite Integration of an Even Function Using the Properties of Definite Integration

Video Transcript

The function 𝑓 is even, continuous on the closed interval negative eight to eight, and satisfies the definite integral between negative eight and eight of 𝑓 of 𝑥 with respect to 𝑥 is equal to 19 and the definite integral between zero and four of 𝑓 of 𝑥 with respect to 𝑥 is equal to 13. Determine the definite integral between negative eight and negative four of 𝑓 of 𝑥 with respect to 𝑥.

We begin by recalling the property of the integral of an even function That is, the definite integral between negative 𝑎 and 𝑎 of that even function is equal to two times the definite integral between zero and 𝑎 of 𝑓 of 𝑥 with respect to 𝑥. Now, in fact, we’re looking to find the definite integral between negative eight and negative four of our even function. So we’re going to do this in two parts. Firstly, we’re going to split it up and say that the definite integral must be equal to the integral between negative eight and zero minus the integral between negative four and zero. Now, actually, we’ll form an equation using the first part of this integral and the fact that the function is even.

The definite integral between negative eight and eight of 𝑓 of 𝑥 with respect to 𝑥 must be two times the definite integral between zero and eight of 𝑓of 𝑥 with respect to 𝑥. Now, it also follows that this must also be equal then to two times the definite integral between negative eight and zero of 𝑓 of 𝑥 with respect to 𝑥. Of course, in the question, we were told that the definite integral between negative eight and eight of 𝑓 of 𝑥 is 19. So we set 19 equal to two times the definite integral that we’re looking for. And then, we divide both sides of our equation by two. And we see that this is equal to 19 over two. The integral we’re looking for then is equal to 19 over two minus the definite integral between negative four and zero of 𝑓 of 𝑥 with respect to 𝑥.

Now, once again, the function is even. So this must, in turn, be equal to 19 over two minus the definite integral between zero and four of 𝑓 of 𝑥 with respect to 𝑥. Remember, this is because even functions have reflection or symmetry about the 𝑦-axis. Now, we’re told in the question that this definite integral is equal to 13. Then to evaluate 19 over two minus 13, we write 13 as 26 over two. So we’re looking to find 19 over two minus 26 over two which is negative seven over two. And so, we found the definite integral between negative eight and negative four of 𝑓 of 𝑥 with respect to 𝑥. It’s negative seven over two.