Suppose the region bounded by the curve y=f(x) from x=0 to x=4...

Question

Suppose the region bounded by the curve y=f(x) from x=0 to x=4 (see figure) is revolved about the x-axis to form a solid of revolution. Use left, right, and midpoint Riemann sums, with n=4 subintervals of equal length, to estimate the volume of the solid of revolution.

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Accepted Answer

Hello. In this video, we are given the region under Y is equal to F of X from X is equal to 2 to X is equal to 6, and we are told that this is being revolved about the x axis. By using the right to rhyme in sum with N equal to 4 subintervals, we want to estimate the volume of the solid of the revolution. OK, so let's go ahead and list out something important here. First, we are told that we are revolving this given shape about the X axis. Now, if we were asked to find the volume of the shape when it's revolved about the X axis, we can approach this problem by using the disk method. By using the disk method, the volume is defined as the following. The volume is equal to pi multiplied by the definite integral from A to B. Of a squared function F of XD X. But in this case here, what we want to do is we want to estimate this volume by using a rightyman sum. So what we want to do is you want to go ahead and express this integral as a summation. So we can rewrite this integral as a summation to be pi multiplied by the summation from 1 ton of F of X sub I multiplied by delta X. So, here, Delta X. Is defined as the difference of our boundaries divided by the amount of subintervals we are working with. Now the problem tells us that we are working on the values from 2 to 6, so we can go ahead and define delta X. And one more thing to note is that the problem is telling us to use 4 equal sub-intervals. So, let's go ahead and first define delta X for the summation. Delta X will equal to the larger interval minus the smaller one, which is 6 minus 2, divided by the amount of sub-intervals we are working with, which is 4. This will simplified to be 4 divided by 4, which is equal to 1. So this is going to be the value of delta X for summation. Now, again, we want to approximate this volume by using right Riemann sums. And so what this means is that when we split this area from 2 to 6 into 4 equal areas, that means that each area will have a width of 1. Now, those endpoints are going to be 345, and 6. So, what this means is that if we were to expand this summation that we came up with for the volume, we can rewrite the summation. To be the summation or pie. Multiplied by Delta X equal, not equal, multiplied by the quantity. F of 3. Plus F of 4. Plus 5 of 5 plus F of 6. Now what we want to do in this case is we want to go ahead and find the values of the functions. Of the function at each of these endpoints. And one more thing to note too, keep in mind that the original function in our summation is being squared, and that means that each value that we find is also going to be squared as well. So starting at 3, the value of the function of 3 is going to be 3. The value of the function of 4 will be 5, the value of the function of 5 will be 6, and the value of the function at 6 will be 8. So by substituting in these values, we will have i multiplied by a value of delta X, which is 1. Multiplied by 3 squared. Plus 5 squared. Plus 6 squared. Plus 8 squared. Now all we need to do is we need to simplify. This is going to simplify to be pi multiplied by the quantity, 9 + 25 + 36 + 64. And this is going to sum to give us a final value of 134 i. So this is going to be the volume estimation by using rightyman sums. And the overall solution to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Suppose the region bounded by the curve y=f(x) from x=0 to x=4 (see figure) is revolved about the x-axis to form a solid of revolution. Use left, right, and midpoint Riemann sums, with n=4 subintervals of equal length, to estimate the volume of the solid of revolution.

Key Concepts

Solid of Revolution

Riemann Sums (Left, Right, Midpoint)

Volume Approximation Using Disk Method

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