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 = ln(2) about the line x = ln(2).

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. Compute the volume of the solid form by revolving the region in the first quadrant bounded by the coordinate axes. The curve y is equal to e to the power of x divided by 2. And the vertical line X is equal to 2 multiplied by the natural log of 2, about the vertical line, X is equal to 2 multiplied by the natural log of 2. OK, so it appears for this particular prompt, we're asked to take all the information that is provided to us by the prom itself, and we're asked to compute what the volume of the solid is that is formed by revolving this specific region in the first quadrant that is bounded by the axis and the curve Y and our vertical lines for both X is equal to 2 multiplied by the natural log of 2. And X is equal to 2 multiplied by the natural log of 2. So With that in mind, now that we know that we're ultimately trying to solve for the volume of this particular solid, our first step that we have to take is based on our prior knowledge. 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 that means based on our prior knowledge, we should note that the volume V will be equal to 2 pi multiplied by the integral evaluated from A to B of the radius R multiplied by the height HDX, where the radius R will be equal to the absolute value of X minus C. And the height h will be equal to fx minus g x. So let us state for this particular problem that F of X will be equal to e to the power of x divided by 2, and that g of x will be equal to 0, and this is once again for the x axis, so we don't get confused here. And we need to note that A will be equal to 0, and this is for the Y axis. And finally, we need to note that B will be equal to 2 multiplied by the natural log of 2. So now at this point, we need to we are going to need to determine what the radius and the height are. So the radius R will be equal to the absolute value of x minus 2 multiplied by the natural log of 2, which will be equal to 2 multiplied by the natural log of 2 minus X, and it'll be on the interval square brackets of 0.2 multiplied by the natural log of 2. Where once again square brackets indicate a closed interval and our heighth will be equal to e to the power of x divided by 2. So now we need to substitute our values back into our formula. And when we do, we'll be able to write that the volume V will be equal to 2 pi multiplied by the integral evaluated from 0 to 2 multiplied by the natural log of 2 of parentheses of 2 multiplied by the natural log of 2 minus x multiplied by e to the power of x divided by 2. The x and when we simplify our volume expression to its most simplified form, we'll be able to simply write. So I'm going to skip a few steps, but you'll just use algebra to manipulate it to a simplified expression. You'll find that V is equal to 2 pi multiplied by square brackets of 2 multiplied by the natural log of 2 multiplied by the integral, evaluate from 0 to 2 multiplied by the natural log. Of 2 of e to the power of x divided by 2 d x minus the integral evaluated from 0 to 2 multiplied by the natural log of 2 of x multiplied by e to the power of x divided by 2 d x fantastic. So, moving right along here, we now need to integrate each term, so focusing our attention first on the integral, evaluate from 0 to 2 multiplied by the natural log of 2 of E to the power of x divided by 2. DX. And this will be equal to 2 multiplied by the integral evaluate from 0 to 2 multiplied by the natural log of 2. Of e to the power of x divided by 2, d of x divided by 2, which will be equal to 2 multiplied by e to the power of x divided by 2, evaluate from 0 to 2 multiplied by the natural log of 2. Which will be equal to 2 multiplied by square brackets of e to the power of 2 multiplied by the natural log of 2 divided by 2 minus e to the power of 0 divided by 2. Which when we simplify this expression, we will find that it's simply equal to 2. So solving for our second term, noting that we have to recall and use the integration by parts method which states that the integration by parts formula is the integral of u dV is equal to u multiplied by v minus the integral of vduu. And applying our integration by parts method, we could state that you will be equal to x, and that will mean therefore that DU by Dx will be equal to 1, which will mean that DU will be simply equal to dx, and that will mean that we can let dV be equal to e to the power of x divided by 2. D X. So now we need to find v. So we need to take the integral of dv, which is equal to the integral of e to the power of x divided by 2. Dx, which will mean that v will be equal to 2 multiplied by the integral of e to the power of x divided by 2 multiplied by d of x divided by 2, which will be equal to 2 multiplied by e to the power of x divided by 2. And we now need to substitute this back into our formula, so we need to go ahead and write the integral evaluate from 0 to 2 multiplied by the natural log of 2 of x multiplied by e to the power of x divided by 2. DX is equal to 2 multiplied by x multiplied by e to the power of x divided by 2. And we can't forget that this is evaluated from 0 to 2 multiplied by the natural log of 2. -2 multiplied by the integral evaluate from 0 to 2 multiplied by the natural log of 2 of e to the power of x divided by 2 d x. Which will be equal to square brackets of 2 multiplied by X multiplied by e to the power of x divided by 2 minus 2 multiplied by E, so it's gonna be multiplied by 2. Multiplied by e to the power of x divided by 2. Evaluated from 0 to 2 multiplied by the natural log of 2. Which will be Equal to drum roll, this will be equal to square brackets of 2 multiplied by e to the power of x divided by 2 multiplied by parentheses of x minus 2 evaluated from 0 to 2 multiplied by the natural log of 2. And we need to note that at X is equal to 2 multiplied by the natural log of 2, that will mean that 2 multiplied by e to the power of 2 multiplied by the natural log of 2 divided by 2. Multiplied by parentheses of 2 multiplied by the natural log of 2 minus 2 will be equal to simply when we write our final answer is most simplified form 8 multiplied by the natural log of 2 minus 8. And for X is equal to 0, so when X is equal to 0, so meaning we're plugging in a value of 0 in for X, that will mean that 2 multiplied by e to the power of 0 divided by 2 multiplied by parentheses of 0 minus 2 will be simply equal to -4. So that will mean therefore we could write that the integral evaluated from 0 to 2 multiplied by the natural log of 2 of x multiplied by e to the power of x divided by 2. The x will be equal to 8 multiplied by the natural log of 2 minus 8 minus -4, which will be equal to simply 8 multiplied by the natural log of 2 minus 4. So now we officially need to combine our results, so we need to say that the volume V will be equal to 2 pi multiplied by square brackets of 2 multiplied by the natural log of 2 multiplied by 2 minus parentheses of 8 multiplied by the natural log of 2 minus 4. Which when we simplify this expression down to its most simple form, we will find that our final answer without a doubt will be 8 pi multiplied by parentheses of 1 minus the natural log of 2. And since we're dealing with surface, or rather we're dealing with the volume, so surface area, it would be square units, but specifically, since we're solving for volume, we need to make sure our units are in cubic units, because once again, we're dealing with volume, and volume is in units of cubic units. And that's it. That is our final answer. Hooray, we did it. So going back to look at the prom itself, we can safely say that the volume of this particular solid is without a doubt, once again, so the volume of our solid is. A pi multiplied by parentheses of 1 minus the natural log of 2 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 = ln(2) about the line x = ln(2).

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