Finding volume: Find the volume of the solid generated by revo...

Question

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.

a. About the y-axis.

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes. The curve y is equal to e to the power of -2 x, and the line x is equal to 1 about the y axis. OK. So it appears for this particular prompt based on all the information that's provided to us, we're ultimately asked to determine what the volume of the solid is that is generated by revolving the region. In the first quadrant bounded by the coordinate axes and our specific curve y is equal to e to the power of -2x and the line x is equal to 1 when it's revolved about the y axis. So now that we know what we're ultimately trying to solve for. Our first step that we need to take in order to solve this problem is we need to recall and use the shell method formula specifically for cylindrical shells for revolution about a vertical line, X is equal to C. So, as we should recall, based on our prior knowledge, the volume V will be equal to 2 multiplied by pi, multiplied by the integral evaluated from A to B of the radius R multiplied by the height HDX. So I'm using lowercase r to denote radius and lowercase h to denote height. So, we also need to recall that the radius R is equal to the absolute value of x minus C. And the height h is equal to fx minus g x. OK. So, for this particular problem, let us state that FX will be equal toe to the power of -2X, and let us state that G X will be equal to 0, and this is for the x axis specifically. And let us also note that A will be equal to 0, and this is for the Y axis, and let us also note that B will also be equal to 1. OK, so now at this stage, we need to determine what the radius and height is, and we need to know once again based on what the problem tells us that since the region is revolved about X is equal to 0, so let's make a note of that. So, in blue we'll say that it's revolved about X is equal to 0, so I'll put a little R and quotation marks, so we know that it's revolved about X is equal to 0. That will mean that our radius R will be equal to the absolute value of X minus 0, which will be equal to x. And this will be on the interval, specifically square brackets of 0.1, where square brackets indicate a closed interval. So the radius r will be equal to x on the closed interval 0.1, and our heighth will be equal to e to the power of -2x. So now we need to substitute these values into our shell method formula, and when we do that, we will find that the volume B will be equal to 2 pi multiplied by the integral evaluated from 0 to 1 of x multiplied by e to the power of -2 x dx. So now, in order to solve for this integral that we are given, we will need to recall and use the integration by parts method, which as you should recall, the integration by parts formula states that the integral of uDV is equal to u multiplied by v minus the integral of VDU. Fantastic. So with that in mind, let us state that you will be equal to x. Then that will mean that DU by DX will be equal to 1, which will mean that DU will be equal to DX. And let us state that DV will be equal to e to the power of -2 XDX. So now we need to solve for V, so we need to find V. So we need to take the integral of DV. And the integral of dv will be equal to the integral of e to the power of 2 x dx, which will mean that v will be equal to 1/2 multiplied by the integral of e to the power of -2 x d of 2x. Which will be equal to 1/2 multiplied by e to the power of 2x. OK, so now we need to substitute our new values into our formula. So we need to take the integral evaluated from 0 to 1 of x multiplied by e to the power of -2 XDX. Which will be equal to 1/2 multiplied by x multiplied by e to the power of -2x, evaluated from 0 to 1. And then this will be minus 1/2 multiplied by the integral evaluated from 0 to 1 of e to the power of -2 x dx, which this will be all equal to square brackets of 1/2 multiplied by x multiplied by e to the power of 2 x. Plus 1/2 multiplied by parentheses 1/2 multiplied by e to the power of 2x evaluated from 0 to 1. And this will be equal to -1 divided by 2 multiplied by e to the power of 2 x multiplied by square brackets up x plus 1/22, and this will be once again evaluated from 0 to 1. Fantastic. So now we need to consider the condition at x is equal to 1. So we're going to be plugging in a value of 1 in for x. We will get -1 divided by 2 multiplied by e to the 2 multiplied by 1. Multiplied by square brackets of 1 + 1/22, which will be equal to -3 divided by 4 multiplied by e to the 2. And at x is equal to 0, so now we're plugging in a value of 0 in for x, and we're just going to plug it in. So we're going to take 1 divided by 2 multiplied by e to the power of 2 multiplied by 0, multiplied by square brackets of 0 plus 1/2, which will be simply equal to -14. Fantastic. So moving right along here, that will mean that we can go ahead and write that our equation for the volume V will be equal to 2 pi multiplied by square brackets of -3 divided by 4 multiplied by e to the power of 2. Minus 1/4. And what we need to do is we need to take our current expression for the volume, and we need to simplify it down to its most simple form. So, I'll do a few steps. I'll do one step, but then you'll need to finish the rest on your own. So you will find when we start to simplify this expression, V will be equal to 2 pi multiplied by square brackets of -3 divided by 4 multiplied by e to the power of 2 plus 1 divided by 4. And as you could see, we could factor out a 1/4. And when we factor out the 1/4 and we write our final answer in its most simplified form, I'm going to skip a few steps because you're just going to be manipulating it using a little bit of algebra. We will find V, the volume is equal to i divided by 2 multiplied by e to the 2, multiplied by parentheses of e to the 2 minus 3. And since we're dealing with volume, don't forget that our units will be cubic units. And that's it. That is our final answer. Hooray, we did it. So going back to look at the prom itself to recap what we've learned, we could safely say without a doubt that our final answer for the volume of this specific solid will once again be drum roll. I divided by 2 multiplied by e to the 2 multiplied by parentheses of e to the 2 minus 3 cubic units. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.a. About the y-axis.

Watch next

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.
a. About the y-axis.