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

Question

Volumes

Find the volume of the solid generated by revolving the region bounded by the parabola y² = 4x and the line y = x about

d. the line y = 4

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 formed by revolving the region bounded by the parabola. Y 2 is equal to 8 x, and the line y is equal to 2 x about the line y is equal to 6. OK, so it appears for this particular prompt, we're asked to solve for the volume of the solid for this particular prompt 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 look at the information that is provided to us, and based on this information, we need to be able to recognize, based on our prior knowledge that we should be able to recall and use the shell method formula in order to solve for the volume for this particular problem. So we can use the shell method formula, and this is for cylindrical shells that have been revolved about a vertical line using the format X is equal to A. So with that in mind, as we should recall, the formula for the shell method states that the volume V is equal to 2 pi multiplied by the interval evaluated from A to B of parentheses of the radius R multiplied by parentheses of h the height. DY Fantastic. Where the radius R is equal to the absolute value of Y minus A. And the height H is equal to FY minus GY. Fantastic. We need to note that we are given y 2 is equal to 8x, which will mean that x is equal to y2 divided by 8, and we're told that y is equal to 2x, which will mean that X is equal to Y divided by 2. Fantastic. So with this in mind. We now need to find the intersection by solving 4 Y 2 divided by 8 is equal toy divided by 2. So, essentially, what we need to do is we're going to be rearranging this equation or expression in order to isolate and solve for Y. So, when we begin to do that, we will get 2 multiplied by Y to the power 2. Is equal to 8 Y. And eventually we'll be able to factor the expression, so I'm gonna skip a few steps, but you will know that you factored the expression correctly if you get Y multiplied by parentheses of Y minus 4 is equal to 0, which will mean that Y is equal to 0 or Y is equal to 4. Fantastic. So with that in mind, we now need to determine the shell radius. So the radius R is equal to the absolute value of Y minus 6, which will mean that it will be equal to 6 minus y. On Y is an element of the closed interval square brackets of 0.4. Fantastic. That's what that weird looking symb this E looking symbol means is an element of. So, once again, Y is an element of the closed interval square brackets of 0.4. So now with that in mind, we now need to determine the shell height. So we need to note that y divided by 2 is greater than or equal to y to the 2 divided by 8. On Y is an element of the closed interval 0.4. Thus, that will mean that the height H is equal toy divided by 2 minus y to the power of 2. Divided by 8. Which will be all equal to 4 multiplied by y minus y to the power of 2 all divided by 8. So now we need to set up our integral and begin to solve for our final answer for the volume, so we could write that V is equal to 2 pi multiplied by the integral evaluated from 0 to 4 parentheses of 6 minus y multiplied by parentheses of 4 y. Minus y to the 2 divided by 8. DY Which we could go ahead and even further we could write that v the volumes equal to pi divided by 4 multiplied by the integral evaluated from 0 to 4 of parentheses 6 minus y multiplied by. 4 y minus y to the power of 2 Dy. And we can write even more further along. We could write that the volume V is equal to pi divided by 4 multiplied by the integral evaluated from 0 to 4 of 24, so it's gonna be in parentheses of 24 y minus 10y to the power of 2 plus y to the power of 3. DY. And now we can slowly begin to perform our evaluation by writing the volume V is equal to pi divided by 4 multiplied by square brackets of 24 divided by 2 multiplied by y to the 2 minus 10 divided by 3 multiplied by y to the power of 3. Plus 1 divided by 4 multiplied by y to the power of 4, all in square brackets evaluated from 0 to 4. Which will mean that v is equal to i divided by 4 multiplied by square brackets of 12 multiplied by y to the 2 minus 10 divided by 3 multiplied by y to the power of 3. Plus 1 divided by 4 multiplied by y to the power of 4 in square brackets evaluated from 0 to 4. So that will mean that we could go ahead and write that the volume V is equal to pi divided by 4 multiplied by square brackets of parentheses of 12 multiplied by 42 minus 10 divided by 3 multiplied by parentheses of 4 to 3. Plus 1 divided by 4 multiplied by 4 to the power of 4. In parentheses minus parentheses of 12 multiplied by 0 to the power of 2. -10 divided by 3 multiplied by 0 to the power of 3. Plus 1 divided by 4 multiplied by 0 to the power of 4. Which when we begin to simplify, we will find that v is equal to pi divided by 4 multiplied by square brackets of 12 multiplied by 16 minus 10 divided by 3 multiplied by 64 plus 64. And when we simplify this expression down to its most simple form, so I'm going to skip a few steps, but essentially we need to keep manipulating our expression until we can't manipulate it anymore. So we will find that the volume V will be equal to 32 pi divided by 3, and don't forget that our units will be cubic units. And that's it. We've officially solved for this problem. Hooray, we did it. So going back to look at the prom itself, we will determine that once again our final answer without a doubt for this particular solid will be drum roll. 32 pi divided by 3 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 by the parabola y² = 4x and the line y = x aboutd. the line y = 4

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Volumes
Find the volume of the solid generated by revolving the region bounded by the parabola y² = 4x and the line y = x about
d. the line y = 4