39–44. Shell method about other lines Let R be the region boun...

Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

x =2

x =2

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says let T be the region bounded by Y is equal to X2, X is equal to 2, and Y is equal to 0. Find the volume of the solid form when T is revolved about the line X is equal to 4 using the shell method. So the first thing we want to do is create a sketch of our curves. And so here we're going to draw X and Y axis focusing on the first quadrant. The first curve that we want to draw is Y is equal to X2. So that's just gonna be a parabola, and here we're just going to focus on the first quadrant. So that's going to start at the origin and just increase upwards. So we'll label that as Y is equal to X2. The next curve that we want to draw is X is equal to 2, so this is going to be a vertical line at X equal to 2. And for the final line, that's going to be Y is equal to 0, and that corresponds to our x axis. So we'll just label that as Y is equal to 0. And so the region that is bounded by these three curves is the region that we're going to be revolving about the line X equal to 4, which we will draw as a dashed line on our graph and label it as X is equal to 4. Now, since we're revolving about a vertical line using the shell method, that means we're going to be integrating with respect to X. So that means that our limits of integration are going to go from 0 to 2. We also need to determine the radius and height of each of our shells. So, for our radius, that is just going to be the distance from the line that we're revolving about to the location of our shell. Now, since we're revolving about the line, X is equal to 4, that means our radius is just going to be equal to 4 minus X. Now for our height, we call that our height, it's just going to be equal to the difference of the Y values of the curves that bound our region from above to that that bounds our region from below. So the curve that bounds our region from above is going to be Y equal to x2. So we'll have X2 minus the curve that bounds our region from below, which is Y is equal to 0. So we'll have 0. So that means our height is just going to be equal to X2. So now we can set up the interval for our volume. So recall that using the shell method that our volume V is going to be equal to 2 pi, multiplied by the interval from 0 to 2 of our radius, which is going to be the quantity of 4 minus X in quantity, multiplied by our height, which is X squared, and that gets multiplied by DX since we're integrating with respect to X. So to perform this integration, first we'll multiply everything out inside the integram. In doing so, we see that this is going to be equal to 2 pi, multiplied by the integral from 0 to 2 of the quantity of 4 X squad minus X cubed in quantity DX. So now we can integrate term by term. In doing so, we see that our volume is going to be equal to 2 pi multiplied by the quantity of 4, multiplied by the interval of X with respect to X, and when we integrate X word with respect to X, we get X cubed divided by 3. Then we'll have minus the integral of X cubed with respect to X, and we perform that integration, we get X to the 4th divided by 4 in quantity, and this needs to be evaluated from 0 to 2. 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 4 divided by 3, multiplied by 2 cubed. Minus 24th divided by 4. Minus 4 divided by 3, multiplied by 0. Q. Plus 0 4th power divided by 4 in quantity. So simplifying this expression, this is going to be equal to 2 pi, multiplied by the quantity of. 32 divided by 3. -4. 0 + 0 in quantity. And simplifying this this expression, we see that this is going to be equal to. 2 i multiplied by 20 divided by 3. And evaluating this expression, we see that our volume V is going to be equal to. 40 pi divided by 3. Cubic Units. So that is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

x =2

Key Concepts

Shell Method for Volume

Setting up the Shell Radius and Height

Limits of Integration

Watch next

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.x =2

Key Concepts

Shell Method for Volume

Setting up the Shell Radius and Height

Limits of Integration

Watch next

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

x =2