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

Question

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

d. the line y = 2

Accepted Answer

Find the volume of the solid obtained by revolving the region enclosed by Y equals 2 X minus X2 and the x axis about the line Y equals 4. We have 4 possible answers, being 96 pi divided by 5, 48 pi divided by 5, 64 pi divided by 15, or 64 pi divided by 5, all in cubic units. Now to solve this, we will use the washer method. This formula is given by V equals pi. Multiplied by the integral from A to B. Of Rx 2 minus r x 2 d x Where large R is the outer radius, small r is the inner radius. Let's first find our bounds. We can do this by setting 2 x minus x2 equals to the x axis, which is just 0. 2 X minus X2 equals 0 gives us X2 equals 2 X or X equals 2. It also gives us X equals 0, meaning our bounds go from 0 to 2. Now, our larger radius, in this case, is the line Y equals 4. So, we will say, Or Squared Minus Our inner radius Now the inner radius will be 4 minus 2X minus X2. This is just for minus. 2 x minus x2, which changes the sign to x2. Now from here, we just have a bunch of algebra to simplify. We have pi integral from 0 to 2 of 16. Minus, now that was squared, because we have to square every term and factor. X 4 minus 4 X 3rd. Plus 12 x 2 -16 x. Plus 16 we would find that by foiling. Now we can simplify this even further. Pi integral from 0 to 2. And now we'll distribute our negative. We get g x 4. Plus 4 x to the 3rd. -12 x 2. Plus 16 x. Now notice here, we had 16 minus and 16 in the parentheses. The 16s will go away. Now we can take our integral. We'll use the power rule Which gives us pi multiplied by g x 5 divided by 5. Plus 4 x 4 divided by 4. -12 x 3 divided by 3. Plus 16 x squared divided by 2, from 0 to 2. Now plugging in 0 for all of these will just give us 0, so we just need to plug in 2. We get paid. Multiplied B 48 divided by 5. We would get that by plugging in 2 to every x value. We then get 48 pi divided by 5 cubic units as our volume. This is the answer to our problem, and this corresponds with answer B. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

VolumesFind the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x aboutd. the line y = 2

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Volumes
Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about
d. the line y = 2