Let R be the region bounded by the following curves. Find the ...

Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=4−x^2,x=2, and y=4; about the y-axis

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says let S be the region bounded by the curves. Y is equal to 9 minus X squad, X is equal to 3, and Y is equal to 9. Find the volume of the solid generated when S is revolved about the y axis, and we're given 4 possible choices as our answers. For choice A, we have 7 pi divided by 2. For choice B, we have 81 pi divided by 2. For choice C, we have 45 pi divided by 7, and for choice D, we have 26 pi divided by 6. So, the first thing we want to do is to draw a sketch of our curves on a graph so that we can identify our regions. So we'll draw our X and Y axis so that we focus on the first quadrant. And the first curve that we want to plot is Y is equal to 9 minus X2. So this is a problem that is concave down that has a vertex at 0.9. But we also need to have our X intercepts if we want to create an accurate sketch. So to find the X intercepts, we'll set Y equal to 0 and solve for X. So we'll have 0 is equal to 9 minus X2. To solve for X, we'll add X2 to both sides of the equation. So we'll have X2 is equal to 9, and then taking the square root of both sides, we'll have that X is equal to plus or minus 3. And since we're in the first quadrant, we'll just use the positive root. So we're going to have a problem. That is called Cape Town. That begins on the Y axis at Y is equal to 9, and is concave down, and it intercepts the x axis at X is equal to 3. The next curve that we want to draw is X equal to 3, and so that is going to be a vertical line. Add x equal to 3. And then the next curve is Y is equal to 9. So that is going to be a horizontal line at Y is equal to 9. And so the region that is bounded by those three curves is the reason that we're going to be revolving about the y axis. So we're going to be using the washer method in order to find the volume. So recall your formula for the volume used in the washer method. So that's going to be V, it's going to be equal to pi, multiplied by the integral from A to B of the quantity of our outer radius squared minus our inner radius squared in quantity D Y. So here, since we're evolving about the y axis, we need to integrate with respect to Y. That means that we want to write a curves as functions of y. So the only curve that we need to rewrite is Y is equal to 9 minus X2. And so we need to solve this for X, so we can add X2 to both sides and subtract Y from both sides. In doing so, we will get X squared is equal to 9 minus Y and then taking the square root of both sides, we'll get that X is equal to the square root of the quantity of 9 minus Y. And here we've just taken the positive square root since we're in the first quadrant. So now we just need to identify our inner and outer radius. So, for our outer radius, which we'll denote as our out, That is just going to be equal to the curve that is bending our region on the right, and that is X is equal to 3, so that means that our out is equal to 3. And if we look at our inner radius, which we'll denote as RN, that is going to be equal to the curve that is bounding our region from the left, and that is X is equal to the square root of the quantity of 9 minus Y. So that means that RN is going to be equal to the square root of the quantity of 9 minus Y in quantity. So now we can substitute these results back into our volume formula. So volume V is going to be equal to pi, multiplied by the integral, and here our limits of integration are going to go from 0 to 9, since that is the Y values that our region spans over. Then for the integram, we will have the quantity of Rals squared, which is going to be 32 minus RN2, which is going to be the quantity of 9 minus Y in quantity, D Y. So now we just need to simplify our integrat. So doing so, this is going to be equal to pi, multiplied by the integral from 0 to 9, and inside the integram we will have the quantity of 9 minus 9 plus Y in quantity DY and so find further, this is going to be equal to pi multiplied by the integral from 0 to 9 of Y D Y. So now we can actually perform the integration. So for the integration, this is going to be equal to pi, multiplied by, and here we're going to integrate Y with respect to Y and recall that when we integrate Y with respect to Y, that we get Y quad divided by 2, and this needs to be evaluated at 0 to 9. And so substituting in our limits of integration, this is going to be equal to pi multiplied by the quantity of 92 divided by 2 minus 02 divided by 2 in quantity. And if I weight in this question, this is going to be equal to 81 pi divided by 2. 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.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=4−x^2,x=2, and y=4; about the y-axis

Key Concepts

Setting up the region bounded by curves

Volume of solids of revolution about the y-axis

Using the shell method for volume calculation

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