Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = e^x $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $ ; about the x-axis

$V=\pi\left[\frac{e^{2}}{2}-\frac{e^{-2}}{2}\right] \approx 11.39$

Applications of Integration

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here we want to find the volume of a solid obtained by protein. That region bounded by y equals zero X equals negative one X equals one And why equals E to the X? We're rotating about the X axis, So let's first begin by drawing out the region. Here's the X and Y axis. The Y equals zero line is basically just the X axis so we can signify the X equals negative one. We could mark this as a negative one. We're here. So it would just be a vertical line. And the X equals one would just be a vertical line where X equals one now for y equals e to the X. It grows exponentially like that where that's X equals zero. Why equals one? Because eat, the zero is just one. So now that you got the region bounded, you can see that this area covered in red is the region, and we are rotating this about the X axis. Now, when we wrote about the X axis, well, essentially we're doing is that creating multiple circles of this radius and stacking them on top of each other, or, in this case, next to each other. So we need first find the cross sectional area in terms of X because as exchanges, the cross sectional area also changes. So we can just call this a function of X, and that's equal to pi r squared because, as I said before, we're making circles when we're rotating. Are in this case is just the white where the hunt, so in this case are is equal to y is equal to eat of X. So when we played that in, we get a's X is equal to you pie in two to six. So now we have the cross sectional area function. We couldn't go ahead with our interval to find the body we're integrating from X equals native one to one. And we can just put what we previously found as the cross sectional area from me within the interval so we can put pie into two acts. The ex. It's a pie we can pull out because there's no relation to X. So the answer derivative of eating two X is just eat to the two X over two because you need taking to account that the two that comes when you do the changeable so you can just say this is equal to pi times Easy to ACS over to nine one one. So, really, there's no quick way to simplify this on. What we end up doing is that we just plug in one on eight of one. So you get pie times. He too, the two it, too, because when he plumb in X equals one over two minus E to the one you cooking X equals negative one. You get negative two for two, and this is in essence, the final answer. You can plug this into a calculator, and if you do, you will get the answer. Eleven for meat. Oh three nine. It's so either this woodwork, or you could also submit this as an answer as well.