Finding volume
The region in the first quadrant that is ...

Question

Finding volume

The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

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. Consider the region in the first quadrant bounded by the curve Y is equal to 4 x multiplied by the square root of 2 minus x. And the x axis from x is equal to 0 to the point where the curve meets the axis. This region is revolved about the y axis to form a solid. Compute the exact volume of that solid, OK. So it appears for this particular prompt, based on all the information that is provided to us, we are asked to compute what the exact volume of this specific solid is. So ultimately, once again, we're asked to solve for the exact volume value. So once again, we're solving for the volume of this solid. So now that we know what we're ultimately trying to solve for, our first step that we need to take in order to solve for this problem is we need to find the bounds of integration. So the problem itself tells us that y is equal to 4 x multiplied by the square root of 2 minus x. And that the region is bounded by X is equal to 0 at the point where Y is equal to 0. Other than X is equal to 0. So with that in mind, we need to set Y is equal to 0, and when we do, we will find that 4 x multiplied by the square root of 2 minus X will be equal to 0. And we need to note that this will be 0 when X is equal to 0, or when the square root of 2 minus X is equal to 0, which will mean that X is simply equal to 2. Fantastic. So that means, undoubtedly, our bounds are X is equal to 02 X is equal to 2. Fantastic. So, our 2nd step now, moving on to step 2, is we need to set up the volume integral by recalling and using the shell method. So once again, we need to revolve about the Y-axis and using the formula for the shell method focusing on a vertical strip, we can state that the equation will be the volume V. Will be equal to 2 pi multiplied by the integral evaluated from A to B of x multiplied by y dx, and we need to note once again for this case y is equal to 4 x multiplied by the square root of 2 minus x. So that will mean we could write that the volume V will be equal to 2 pi multiplied by the integral evaluated from 0 to 2 of x multiplied by 4 x multiplied by the square root of 2 minus xDX. Which will be equal to 8 pi multiplied by the integral evaluated from 0 to 2 of x to the power of 2 multiplied by the square root of 2 minus xDX. Fantastic. Our third step now is we need to compute the integral, so we need to evaluate i is equal to the integral evaluated from 0 to 2 x 2 multiplied by the square root of 2 minus xdx. And we will need to use. The, so we're gonna be using integration by parts and a little bit of substitution. So let us note that we will state that U is equal to 2 minus X, which will mean that DU. Will be equal to DX, and this is when X is equal to 0, that will mean U is equal to 2, and when X is equal to 2, that will mean that you will be equal to 0. Fantastic. So, that means that X is equal to 2 minus U will mean that X 2 is equal to 2 minus U 2. So, with that in mind, we can go ahead and write that I is equal to the integral evaluate from x is equal to 0 to x is equal to 2 of x 2 multiplied by the square root of 2 minus xdx. Which will be equal to the integral evaluated from u is equal to 2 to u is equal to 0 of parentheses of 2 minus u. To the power of 2 multiplied by the square root of u multiplied by parentheses of negative duu. So now we need to switch the limits and remove the negative symbol, and when we do that, we will find that i is equal to the integral evaluate from u is equal to 0 to u is equal to 2 parentheses of 2 minus u to the 2 multiplied by the square root of u. Do you? So now we need to expand, so we need to take parentheses of 2 minus u 2, which will be equal to 4 minus 4U plus u 2. So, that will mean that I will be equal to the integral evaluated from 0 to 2 of parentheses or multiplied by the square root of u. -4 u multiplied by the square root of u plus u to the power of 2 multiplied by the square root of u. Do you OK. So, we now need to integrate each term individually. So we need to take the integral of 0 to 2, so once again, the integral evaluated from 0 to 2 of 4 multiplied by the square root of UDU. Which will be Equal to, so I'm gonna skip a few steps because we're integrating each term. So when we integrate the integral evaluated from 0 to 2 of or multiplied by the square root of u DU, we will find that it's simply equal to in its most simplified form, 16 divided by 3 multiplied by the square root of 2. And the integral evaluated from 0 to 2 of 4U multiplied by the square root of UDU will be equal to 32 divided by 5 multiplied by the square root of 2. And finally, the integral evaluated from 0 to 2 of U to the power of 2 multiplied by the square root of uDU will be equal to 16 divided by 7 multiplied by the square root of 2. Which will mean therefore that i is equal to 16 divided by 3 multiplied by the square root of 2 minus 32 divided by 5 multiplied by the square root of 2 plus 16 divided by 7 multiplied by the square root of 2, which when we simplify this expression we will find that it's equal to the square root of 2 multiplied by parentheses of. 128 divided by 105. So therefore, that will mean that our volume V will be equal to 8 pi multiplied by i, which will be equal to 8 pi multiplied by the square 2 multiplied by parentheses of 128 divided by 105. Which will be equal to 1,024 pi multiplied by the square root of 2 divided by 105, and that is our correct and final answer. Hooray, we did it. So that will mean looking at the problem itself we can safely say once again that our final answer for the volume is once again. Drum roll. 1,024 pi multiplied by the square root of 2 divided by 105, and very importantly, we can't forget our units and our units since we're dealing with volume is cubic units. So that's our final answer. 1,024 pi multiplied by the square root of 2 divided by 105 cubic units. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

Finding volumeThe region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

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Finding volume
The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.