Determine the definite integral that describes the shaded...

Determine the definite integral that describes the shaded region above. \int_{-6}^{9} 7du \int_{-6}^{9} 7dw \int_{-6}^{9} 7dw \int_{-6}^{9} 7df \int_{-6}^{9} 7dw \int_{-6}^{9} 7 Hint: Recall that \int_{a}^{b} f(x) dx represents the net area of the region bounded by $f(x)$, $x = a$, $x = b$, and $y = 0$. Determine the value of the definite integral that describes the shaded region above.

Transcript

00:01 So for this question, we're given two functions here.

00:04 We're given x, x equals y squared minus eight.

00:15 And we're given x equals e to the y.

00:22 And we're asked to set up an integral to find this shaded area.

00:26 And this is also between y equals negative one and y equals one.

00:36 Okay, since these are functions, x or functions of y, let's set up an integral on d ,y.

00:42 And our y is going to go from negative 1 to 1.

00:46 So this is going to be an integral, actually.

00:50 Let me come over here to have more room to work.

00:52 The integral from negative 1 to 1.

00:57 And the rightmost function is this e to the y.

01:02 So i have e to the y minus the leftmost function.

01:08 And i'll have to put that in parentheses as well.

01:10 So y squared minus eight and then d y.

01:20 Okay, so we can see that here.

01:22 And i'm going to go ahead and scroll a little bit to create myself a little bit more room.

01:26 So this is going to be the integral of negative one to one of e to the y minus y squared plus 8, dy.

01:47 And then since everything is addition and subtraction and the integral, we can evaluate each one separately.

01:52 So the derivative, or sorry, the integral, well, the derivative, e the y is e the y.

01:56 Therefore, the integral of each of the y is e to the y.

01:58 So that's e to the y.

02:02 And remember the integral of y squared, y is going to go up to the square, the square is going to go up to a cube.

02:09 But then i need to have this one third in front...

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