Volumes
Find the volume of the solid generated by revolv...

Question

Volumes

Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about

c. the line x = 5

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. Calculate the volume of the solid obtained by revolving the region bounded on the left by x is equal to y2 + 2 and on the right by x is equal to 6 about the line x is equal to 6. OK. So it appears for this particular prompt, we're asked to calculate the volume of the solid based on all the information that is provided to us by the prom itself. So, now that we know that we're ultimately solving for the volume of this particular solid, we need to be able to recognize, based on our prior knowledge, that based on the information that's provided to us, we can recall and use the DISC method formula to solve for this particular volume. So, as we should recall, the DISC method formula states that the volume V is equal to pi multiplied by the integral evaluated from A to B parentheses R Y to the power of 2DY. Where we need to note that capital R Y is the radius of the disk. So, at this point, we now need to find the intersection of X is equal to y2 + 2, and X is equal to 6. So that will mean that we can write that Y2 + 2 is equal to 6 because we need to set these two expressions equal to each other. So Y to the power of 2 + 2 is equal to 6. And now we need to use algebra to rearrange this expression, to isolate and solve for Y, and when we do, we will find that Y is simply equal to plus or minus 2. Fantastic. So, with that said, we now need to consider the act of revolving our solid or this disk about the X equals 6. Which gives us our disc of radius. So once again, when we revolve our solid, we have to be specific here, so it's a little bit difficult to visualize at first. So when we take our solid. And we revolve it around our X equals 6 line. That will give us a disk of a radius. R Y is equal to 6 minus y2 + 2, which will be equal to 4 minus y2. And this is 4. Note -2 is less than or equal to y and y is less than or equal to 2. Fantastic. So now, we need to start to solve for our volume. So the volume V is equal to pi multiplied by the integral evaluated from -2 to 2 or minus Y 2. In parentheses to the power 2 Dy, which will be equal to pi multiplied by the integral evaluated from 2 to 2 of 16 minus 8 multiplied by y2 plus y 4. DY And we need to note that our integrand, so let's make another note off to the side here, that our integrand is even, and because our integrand is even, we can therefore write that the volume V is equal to. 2 pi multiplied by the integral evaluated from 0 to 2 of 16 minus 8 multiplied by y2 plus y 4. DY Which means we could write that V is equal to 2 pi multiplied by square brackets of 16 y minus 8 divided by 3 multiplied by y to the power of 3. Plus 1 divided by 5 multiplied by y to the power of 5. Evaluated from 0 to 2. So we could go ahead and write that V is equal to 2 multiplied by square brackets of parentheses of. 16 multiplied by 2. -8 divided by 3 multiplied by 2 to the power of 3. Plus 1 divided by 5 multiplied by 2 to the power of 5. Minus parentheses of drum roll. 16 multiplied by 0 minus 8 divided by 3 multiplied by 0 to the power of 3. Plus 1 divided by 5 multiplied by 0 to the power of 5. Which, when we simplify. This expression skipping a few steps, we will find that V. Is going to be equal to 2 pi multiplied by square brackets of 32 minus 64 divided by 3. Plus 32 divided by 5. And when we simplify this expression down to its most simplified form, skipping another few steps, using it just we're basically manipulating it until we can't simplify it anymore, and we will find that it's equal to 512 pi divided by 15, and we need to note that we can't forget our units, and our units will be cubic units because we are dealing with the volume. And that's it, we've officially solved for this problem. Hooray, we did it. So going back to look at the problem itself, we could say without a doubt that the volume of the solid is once again. 512 pi divided by 15 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.

VolumesFind the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 aboutc. the line x = 5

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Volumes
Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about
c. the line x = 5