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

b. the line y = ―1

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 obtained by rotating the region bounded by the x axis, and y is equal to x 2 minus 6 x from x is equal to 0 to x is equal to 6 about the line y is equal to -9. OK. So it appears for this particular problem, we're ultimately asked to determine what the volume is for this specific solid based on all the information that is provided to us by the problem itself. So now that we know that we're ultimately trying to solve for the volume, our first step that we need to take in order to solve this problem. Is based on the information that's provided to us, we need to be able to recognize that we can recall and use the washer method. And the general formula for the volume when rotating about the x axis is the volume V is equal to pi multiplied by the integral evaluated from A to B of parentheses of square brackets of R X to the power of 2. Minus square brackets of lowercase r x to the power of 2. DX. So we need to note that it's very important that capital R X denotes the outer radius, whereas lowercase r denotes the inner radius. So once again, the volume V is equal to pi. Multiplied by the integral, evaluate from A to B of parentheses of square brackets of R X to the 2 minus square brackets of lowercase r, so where before it's capital R lowercase r x to the 2 dx. And we need to note that the x-axis. So once again, we have the X axis as Y is equal to 0. So, that means that, so let's make a note that 0 is greater than or equal to x 2 minus 6 X. On, since we need to note that this will be on our closed interval, the square brackets indicate a closed interval, square brackets of 0.6, so the closed interval of 0.6. Thus, the region bounded above by Y equals 0 and below by Y equals x 2 minus 6 X on the closed interval of 0.6. So now we need to compute the outer radius, R, as the distance from Y is equal to 0 to Y is equal to -9. Giving R is equal to 0 minus -9, which will be equal to 9. And we also need to compute the inner radius lowercase r x as the distance from y is equal to x 2 minus 6 x toy is equal to -9. So we can go ahead and write that lowercase r x is equal to x to the 2. -6 X minus 9, which will be equal to x 2, minus 6X + 9, which will be equal to x minus 32. So now at this stage, we can recall and apply the Washer formula, which states that the volume V is equal to pi multiplied by the integral evaluated from 0 to 6 of square brackets of 92 minus entheses x minus 32, all the power of 2. DX, which will be equal to. High multiplied by the integral evaluated from 0 to 6 of square brackets of. 81. Minus parentheses of x minus 3 to the power of 4. Don't forget your square bracket, DX. OK, we are on a roll here. So that will mean, so that will mean once again that this will be equal to, so equal to drum roll, pi multiplied by square brackets of. 81 multiplied by the integral evaluated from 0 to 6 of dx minus the integral evaluated from 0 to 6 of parentheses of x minus 3 to the power of 4 dx, and don't forget your square bracket. And at this point, we now need to integrate each term, so focusing on our first term, which is 81 multiplied by the integral evaluated from 0 to 6 of DX, which will be equal to 81 x evaluated from 0 to 6, which will be equal to 81 multiplied by parentheses of 6 minus 0, which will be equal to. 486 focusing on our second term. Which our 2nd integral, we will need to recall and use the substitution method, and we will need to, based on our 2nd integral, we need to let you be equal to x minus 3. That will mean then that DU by DX will be equal to 1, and that will mean that DU will be equal to DX. So, with that said, we now need to rewrite the limits, so that means that X is equal to 0, that will mean that 0 minus 3 will be equal to -3. And that will mean that. X is equal to 6 will mean 6 minus 3 will be equal to 3. So based on our rewritten, so once again we rewrote the limits, so that will mean we could go ahead and write that the integral evaluated from 0 to 6 of entheses of X minus 3 to 4. The X will be equal to the integral evaluated from 3 to 3 of u to the power of 4. DU, which will be equal to 2 multiplied by the integral evaluated from 0 to 3 to 0 to 3. Of You to the power. Of 4 DU. And we need to note that this will be. Equal to 2 divided by 5, so it's gonna be 2 divided by 5. Multiplied by u to the power of 5, evaluated from 0 to 3 of drum roll. Which is going to be equal to 2 divided by 5 multiplied by parentheses of 3 to 5 minus 05. Which when we plug this expression into our calculator and write it in its most simplified form as a fraction, we will find that it's equal to 486 divided by 5. So therefore, we could state that the volume V is equal to pi multiplied by square brackets of 486 minus 486 divided by 5. Which will be equal to pi multiplied by square brackets of 2430 minus 486 divided by 5, which will mean that our volume V is equal to 1,944 pi divided by 5. And don't forget that our units are cubic units because we are dealing with volume. And that's it. That is our final answer. Hooray, we did it. So going back to look at the prom itself. We could safely say that our final answer once again for our volume of this solid, without a doubt will be 1,944 pi divided by 5 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 between the x-axis and curve y = x² ―2x aboutb. the line y = ―1

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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
b. the line y = ―1