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

b. the y-axis

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. Determine the volume when the region bounded by y to the power of 2 is equal to 12 x and y is equal to 3 x is revolved about the y axis. OK. So it appears for this particular poem, we're asked to use all the information that is provided to us by the prom itself in order to help us determine what the volume is of this particular region. That is bounded by y to the 2 is equal to 12 X and Y is equal to 3x when it is revolved about the y axis. So, ultimately, simply put, we are ultimately trying to solve for the volume of this particular problem. So we're trying to solve for the volume. So With that said, based on the information that's provided to us, we need to be able to recognize, based on our prior knowledge, that we can recall and apply the shell method formula, and this is for cylindrical shells for revolutions about a vertical line such as X is equal to A. So that's why we could use the shell method. And as we should recall, the shell method formula states that the volume V is equal to 2 pi multiplied by the integral evaluated from A to B of parentheses of the radius R multiplied by parentheses of H the height DX. So, The volume is equal to 2 pi multiplied by the integral, evaluate from A to B of parentheses R multiplied by parentheses of HDX. Where the radius R is equal to the absolute value of X minus A. And the height H is equal to FX minus G X. Fantastic. So now at this point, we need to find the intersection points by solving y to the power of 2 is equal to 12 x and y is equal to 3 x, and we can solve for these intersection points by setting these two expressions equal to each other, so we need to take. Parentheses of 3 x 2 is equal to 12 x, and what we need to do is we need to take this expression and we need to rearrange it to isolate and solve for x. So when we start to do that, we will get 9 multiplied by x 2 is equal to 12 x. And we will note that we could take it a step further and we could write that 9 multiplied by x 2 minus 12 x is equal to 0, and skipping a few steps, we will find that X will be simply equal to 0. Or X will be equal to 4 divided by 3, because let's note that when we factor out our expression, we will find that the factored out expression is. 3 x multiplied by parentheses of 3 X minus 4 is equal to 0. Fantastic. So with that in mind. We can move on here. We now need to apply our shell method to the region that is bounded by y to the 2 is equal to 12 x, and Y is equal to 3x when it is revolved, which I'll call a capital R in quotes, revolved about the y axis x is equal to 0. So our radius R will be equal to the absolute value of x minus A, where A in this case will be 0, so R is equal to the absolute value of x minus 0, which will be equal to the absolute value of X, which is simply equal to x. On the closed interval of square brackets of 0.4 divided by 3, fantastic. Therefore, the heighth will be equal to 2 multiplied by the square root of 3 x minus 3x. Fantastic. So now we could substitute in our variables into our interval and the volume V. So once again. We could solve for the volume now, so we could plug in our known variables back into our equation for the shell method, and we will find that the volume V is equal to 2 pi multiplied by the integral evaluated from 0 to 4 divided by 3 of x multiplied by parentheses of 2 multiplied by the square root of 3 x minus 3 x. Multiplied by DX. And we can also continue to write that V, the volume, is equal to 2 pi multiplied by the interval, evaluated from 4 divided by 3 of parentheses of 2 multiplied by the square root of 3 multiplied by x to the power of 3 divided by 2 minus 3 multiplied by x 2 DX. Which we could write that V is equal to 2 pi multiplied by parentheses of 2 multiplied by the square root of 3 multiplied by x to the power of 5 divided by 2 divided by 5 divided by 2. And we need to note that we will need to subtract 3 multiplied by x to the 3 divided by 3, and this will be evaluated from 0 to 4 divided by 3. Fantastic, and we could write that V is equal to 2 pi multiplied by parentheses of 4 multiplied by the square root of 3. Divided by 5 multiplied by x 2 multiplied by the square root of x minus x 3, evaluated from 0 to 4 divided by 3. So now we can begin our evaluation. So v is equal to 2 pi multiplied by square brackets of parentheses of 4 multiplied by the square root of 3 divided by 5 multiplied by 4 divided by 3 to the power of 2 multiplied by the square root of 4 divided by 3 minus parentheses of 4 divided by 3. To the power of 3. Minus parentheses. 4 multiplied by the square root of 3 divided by 5 multiplied by 0 to the power of 2. Multiplied by the square root of 0. -03. Fantastic. Which will mean we could write that V is equal to 2 pi multiplied by parentheses of. 4 multiplied by the square root of 3 divided by, so 4 multiplied by the square root of 3 divided by 5, and we need to multiply by 16 divided by 9, multiplied by 2, divided by the square root of 3 minus 64 divided by 27. So what we need to do is we're trying to simplify this expression down to its most simplified form. So you just need to keep manipulating it until you can't manipulate it anymore. And when you do, you will find that your final answer will be equal to 128 high divided by 135. And don't forget that your units will be cubic units. And that's it. That is our final answer. Hooray, we did it. So going back up to the top to look at the prom itself. We could safely say that the volume without a doubt is once again. 128 pi divided by 135 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 aboutb. the y-axis

Watch next

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