9-34. Shell method Let R be the region bounded by the followin...

Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

Graph showing region R bounded by y = √x, y = 0, and x = 4, with a shell method illustration for volume calculation.

y = √x,y=0, and x=4; about the x-axis

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says let R be the region bounded by Y is equal to the square root of X, Y is equal to 0, and X is equal to 9. Find the volume of the solid generated when R is revolved about the X-axis using the shell method. And below the text here, we're given a graph that has the curve Y is equal to the square root of X plotted on it. It also has the curve X equal to 9 plotted on it. We also have our region R shaded on this graph, and this region is bounded above by the curve Y is equal to the square root of X, bounded below by the x-axis, and bended on the right by the curve X equal to 9. We are also given 4 possible choices as our answers. For choice A, we have 91 pi divided by 2 cubic units. For choice B, we have 81 pi divided by 2 cubic units. For choice C, we have 9 pi divided by 2 cubic units. And for choice D, we have 81 pi divided by 4 cubic units. So we're asked to find the volume of solid generated when our region is revolved about the X-axis using the shell method. When revolving a region about the X-axis using the shell method, we're actually going to want to integrate over the Y variable. So, we want to rewrite all of our curves as functions of Y. And the only curve that we really need to rewrite is the curve. Y is equal to the square root of X, and we just need to solve this for X. So squaring both sides of the equation, we will get that X is going to be equal to Y2d. So now we need to set up our integral for finding the volume of the solid generated when our region is revolved about the x-axis using the shell method. So recall that for the shell method, our volume V is going to be equal to 2 pi multiplied by the integral from A to B. Of the radius of our shell. Multiplied by the height of our shell. And we're going to be integrating this with respect to why. So, we just need to identify the radius and height of our shells. So, we're going begin by looking at the radius. And here the radius is just going to be the distance from the X-axis to the location of our shell, and that's just going to be equal to our variable Y. Next, we'll look at the height of our shell. And the height of our shell is going to be distance from X equal to 9 to the curve X equal to Y2. And so that difference is going to give us our height, and so we'll have 9 minus Y2d for our height. So now substituting these results into our formula for the volume, we'll see that our volume V is going to be equal to 2 pi multiplied by the integral, and here, Y is going to go from 0 to 3, since when X is equal to 0, Y is equal to 0, and when X is equal to 9, Y is equal to 3. And then for the integram we will have Y multiplied by the quantity of 9 minus y squad in quantity Dy. So now we just need to perform this integration. So the first step is to distribute our y inside the integram. In doing so, this is going to be equal to 2 pi multiplied by the integral from 0 to 3 of quantity of 9 Y minus y cubed and quantity Dy. So now we can just integrate term by term. In doing so, this is going to be equal to 2 pi, multiplied by the quantity, and when we integrate 9Y with respect to Y, we get 9 Y squared, divided by 2. And then we'll have minus and when we integrate Y cubed, we get to the 4th divided by 4 in quantity, and this needs to be evaluated from 0 to 3. So, substituting in our limits of integration, we see that our volume is going to be equal to 2 pi. Multiplied by the quantity of 9, multiplied by 3 squared, divided by 2, minus 3 raised to the 4th power, divided by 4. Minus 9 multiplied by 0 squared, divided by 2, plus 0 raised to the 4th power divided by 4 in quantity. And simplifying this expression, we see that this is going to be equal to 2 pi, multiplied by the quantity of 81 divided by 2 minus 81 divided by 4 in quantity. And evaluating this expression, we see that our volume is going to be equal to. 81 pi divided by 2. Cubic units. So that is the answer that we are looking for for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = √x,y=0, and x=4; about the x-axis

Key Concepts

Shell Method for Volume

Setting up the Integral with Respect to y

Understanding the Region Bounded by Curves

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