Finding volume
The region in the first quadrant enclosed...

Question

Finding volume

The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

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. Consider the region in the first quadrant bounded by the y axis. The curve y is equal to x multiplied by e to the power of x, and the vertical line x is equal to 1. This region is revolved about the y axis to form a solid. Find the volume of the solid generated by this revolution. OK, so it appears for this particular problem we're asked to take all the information that is provided to us and we're asked to ultimately determine, or rather to find what the volume of this particular solid is that is generated by this revolution. So now that we know that we're ultimately solving for the volume of this solid. Our first step that we need to take is we need to recall how to use the method of cylindrical shells. So as we should recall, a typical shell at. X with. 0 is less than or equal to X and X is less than or equal to 1, will have a radius R, which is X, will have a height. Which is, for this particular problem, Y is equal to x multiplied by E to the power of X, and a thickness, which is DX. So I'll use R to denote radius. So, capital R, which I'll just call R, is X. The height H is Y is equal to X multiplied by E to the power of X. And our thickness, which I'll denote as capital T, is DX. That will mean the volume is dV is equal to 2 pi multiplied by the radius multiplied by the height, multiplied by dx, which will be equal to 2 pi multiplied by x multiplied by parentheses of x multiplied by e to the power of x dx. Which will be equal to 2 pi multiplied by x 2 multiplied by e to the power of xDX. So now we need to integrate from X is equal to 0 to X is equal to 1, and when we do, we will find that the volume V will be equal to the integral of 0 to 1 of 2 pi multiplied by x 2 multiplied by e to the power of x dx. Which will be equal to 2 pi multiplied by the integral evaluated from 0 to 1 of x to the power of 2 multiplied by e to the power of x dx. We now need to compute the integral by using integration by parts, so we need to state that for this prompt you will be equal to x 2. And that DV will be equal to e to the power of XDX. So that will mean that DU will be equal to 2 XDX, and that will mean that V will be simply equal to E to the power of X. So, When we use our integration by parts formula, we will be able to write that the integral of x 2 multiplied by e to the power of x dx will be equal to x 2 multiplied by e to the power of x minus the integral of 2 multiplied by x multiplied by e to the power of x dx. So, 4, let's note 4, the integral of 2 multiplied by x multiplied by e to the power of X D X. Let us state that U is equal to X and that DV is equal to e to the power of XDX. That will mean that DU will be equal to DX. And that will mean that V will be simply equal toe to the power of X. So that means we can go ahead and write that the integral using the integration by parts formula once again we could write that the integral of x multiplied by e to the power of x dx will be equal to x multiplied by e to the power of x minus the integral of e to the power of x dx, which will be equal to x multiplied by e to the power of x minus e to the power of x plus c subscript 1. Where C subscript 1 represents some constant variable. So, that means we could write that the integral of 2 multiplied by x multiplied by e to the power of x dx will be equal to 2 multiplied by parentheses of x multiplied by e to the power of x minus e to the power of x plus C subscript 1. In parentheses. So when we put all of our values together, we'll be able to write that the integral of x 2 multiplied by e to the power of x dx will be equal to x 2 multiplied by e to the power of x minus 2 multiplied by parentheses of x multiplied by e to the power of x minus e to the power of x plus c subscript 1. Which will be all equal to x 2 multiplied by e to the power of x minus 2 multiplied by x multiplied by e to the power of x plus 2 multiplied by e to the power of x plus c subscript 2. Fantastic. So, now at this stage, we need to evaluate from x is equal to 0 to x equal to 1. So when we do that, we will be able to write the integral evaluated from 0 to 1 of x 2 multiplied by e x. DX is equal to parentheses of. X 2 multiplied by e x minus 2 multiplied by x multiplied by e to the power of x plus 2 multiplied by e to the power of x, all evaluated from 0 to 1. So at x is equal to 1, we can go ahead and write 12 multiplied by e to the 1 minus 2 multiplied by 1 multiplied by e to the power of 1, plus 2 multiplied by e to the power of 1, which when we evaluate this expression, we will find that it's simply equal to e. And at x is equal to 0 will be 0 minus 0 + 2 multiplied by e to the power of 0, which is simply equal to 2. So that means that our definite integral is E minus 2. So therefore, we could state that our volume V is equal to 2 pi multiplied by parentheses of E minus 2. And this is our correct and final answer. Hooray, we did it, but we need to make sure we add our units, and we need to note since we're dealing with volume, we have cubic units. And that's it. That is our final answer. Hooray, we did it. So going back to the top to recap what we've learned, we can safely say without a doubt that our final answer for the volume of this particular solid once again is 2 pi multiplied by parentheses of e minus 2 cubic units, and that's it. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

Finding volumeThe region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

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Finding volume
The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.