53–62. Choose your method Let R be the region bounded by the f...

Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x³,y=0, and x=2; about the x-axis

Accepted Answer

Welcome back, everyone. In this problem, let R be the region bounded by the curves Y equals 3 of X cubed, Y equals 0, and X equals 2. Find the volume of the solid generated when R is revolved about the Y axis. A says it's 128 9th of pi cubic units. B says it's 128/7 of pi cubic units. C 63 divided by 128 multiplied by pi cubic units, and D 128 pi divided by 63 cubic units. Now how can we figure out our region are? Well, we can use the disk method. Recall that by the disk method it says that for a solid of revolution above the x axis, the volume is going to be equal to pi multiplied by the integral between the bonds of A and B of the radius squared. OK, with respect to X. So if we can figure out what the radius is for our curve and its bounds, then we can use that to solve for the volume. Do we know both of those? Well, we know that our radius, OK, is equal to Y, which in terms of X will be a third of X cubed. We also know that for our bonds for X, it goes from X equals 0 to X equals 2. So we can say that A equals 0, where and B equals 2. So in that case, then this means that the volume V is going to be equal to pi. Multiplied by the integral between 0 and 2 of X cubed divided by 3 squared with respect to x. Now if we expand our squared term, OK, that's going to become X to the 6th divided by 9. And if we take out our constant here, that means our volume will be 9th of pi multiplied by our definite integral of X to the 6th with respect to X. So all we need to do now is integrate and solve. Now the integral of X to the 6 is a 7th of X to the 7th. So this will be a 7, 9th of pi, sorry. Multiplied by 17th of X to the 7th between the bones of 0 and 2. And now if we apply our bones, this will be a 9th of pi multiplied by 27 minus 0 7th divided by 7. 237 is 128. So this is going to be pi multiplied by 128 or 128 pi divided by 9 multiplied by 7 or 63 cubic units. Thus, the volume of the region are that's bounded by the curves Y equals X cubed divided by 3, Y equals 0, and X equals 2 is 128 pi divided by 63 cubic units. D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x³,y=0, and x=2; about the x-axis

Key Concepts

Volume of Solids of Revolution

Disk and Washer Methods

Setting up Definite Integrals with Given Bounds

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