35–38. Shell and washer methods Let R be the region bounded by...

Question

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis

y = (x−2)³ −2,x=0, and y=25; about the y-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 quantity of X minus 1 in quantity cubed minus 3. X equal to 0, and Y is equal to 5. Find the volume of the solid generated when R is revolved about the y axis using the shell method. So, the first thing we want to do is to create a sketch of our curves so that we can identify our region R. So we'll draw our X and Y axis so that we can focus on the 1st and 4th quadrant. Now, the first curve that we want to draw is Y is equal to the quantity of X minus 1 in quantity cubed minus 3. Now, this is just the curved Y is equal to x cubed, shift to the right 1 unit and down 3 units. And if we set X equal to 0, we see that we have a Y intercept of -4. So starting at our Y intercept of Y is equal to -4, our curve will increase until it reaches the inflex point at X equal to 1, and then our curve will continue to increase, passing through the x axis. And so we'll label this curve on a graph as Y is equal to the quantity of X minus 1 in quantity cubed minus 3. The next curve that we want to draw is X equal to 0, and this just corresponds to our y axis, which we've already drawn. So we'll just label our y axis as X equal to 0. And the final curve that we want to draw is Y is equal to 5. And so this is going to be a horizontal line at Y is equal to 5. And the region that is bounded by all three of those curves is the region that we're going to be revolving about the y axis. And since we're revolving about the y axis using the shell method, we're going to be integrating with respect to X. So we need to determine our limits of integration. So, if we look at the curve that's bounding our region on the left, we see that that's X equal to 0. So that means that our lower limit of integration is going to be equal to 0. And for the upper limit of integration, it's going to be the x value of the intersection point between Y is equal to 5 and Y is equal to the quantity of X minus 1 in quantity cubed minus 3. So to find that intersection point, we'd need to just set the two y values equal to 0 and solve for X. And so we will have that 5 is equal to the quantity of X minus 1 in quantity cubed minus 3. So adding 3 to both sides, we will get that 8 is equal to the quantity of X minus 1 in quantity cubed. Taking the cube root of both sides, we'll get that 2 is equal to X minus 1. And to solve for X, we just need to add 1 to both sides, and we'll see that X is equal to 3. So 3 is going to be the upper limit for our limits of integration. Now, since we're using the shell method, we need to determine our radius and heights of our shell. So we'll first look at the radius of each shell, and our radius is just going to be the distance from the y axis to one of our shells, and this just turns out to be equal to our X coordinate. So our radius is equal to X. Next, we're going to look at the height. And our height here is going to be the distance from the curve that bounds our region from above to the curve that bounds our region from below. So if we look at the curve that's bounding our region from above, that's Y equal to 5. So we'll have 5, and we'll need to subtract from it the curve that bounds our region from below. And so we'll have 5 minus the quantity of the quantity of X minus 1 in quantity cubed minus. 3, since that is the curve that bounds our region from below, and we can simplify this expression, and so that means that our height is going to be equal to 8 minus the quantity of X minus 1 in quantity, cubed. So now we have all the information that we need to calculate our volume using the shell method. So we call your formula for finding the volume using the shell method, that our volume V is going to be equal to 2 multiplied by the interval from 0 to 3 of our radius, which is X, multiplied by our height, which is going to be the quantity of 8 minus the quantity of X minus 1 in quantity cubed in quantity DX. So now we just need to perform this integration. So the first thing we want to do is multiply out that cubic term inside our integral. In doing so, we see that V is going to be equal to 2 pi, multiplied by the integral from 0 to 3 of X, multiplied by the quantity of 8 minus the quantity of X Q. Minus 3 x 2 plus 3X minus 1 in quantity, then another in quantity, then DX. So now we just need to distribute the X and collect like terms. In doing so, we see that our volume is going to be equal to 2 pi, multiplied by the interval from 0 to 3. Of the quantity of 9 X Minus 14th. Plus 3 X cubed. Minus 3 x 2 in quantity d X. So now we just need to integrate term by term. So this is going to be equal to 2 pi, multiplied by the quantity of 9, and here we're going to be multiplying that by the integral of X with respect to X, which gives us X2d divided by 2. Then we'll have minus the integral of X to the 4th with respect to X, and we integrate that, we get X to the 5th. Divided by 5, then we'll have plus 3 multiplied by the interval of X cubed with respect to X, which gives us X to the 4th divided by 4, and then we'll have minus 3. Multiply the interval of X squared with respect to X, which gives us X cubed divided by 3. In quantity, and this needs to be evaluated from 0 to 3. So substituting in our limits of integration, this is going to be equal to 2 pi multiplied by the quantity of 9 divided by 2 multiplied by the quantity of 3 square minus 0 squared in quantity, minus 1 divided by 5 multiplied by the quantity of 3 rates to the 5th power, minus 0 rates to the 5th power in quantity. Plus 3 divided by 4. Multiplied by the quantity of 34 minus 0 power in quantity, then minus the quantity of 3 cubed minus 0 cubed in quantity. So now we just need to simplify this expression. So this is going to be equal to 2 pi, multiplied by the quantity of Anyone Divided by 2. Minus 243 divided by 5. Plus 243 divided by 4. 27 in quantity. Combining everything with a common denominator. This is going to be equal to 2 pi, multiplied by. 513 divided by 20. And then simplified this expression, we see that our volume is going to be equal to. 513 divided by 10. 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.

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis

Key Concepts

Shell Method

Washer Method

Setting up the Region and Limits of Integration

Watch next

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.y = (x−2)³ −2,x=0, and y=25; about the y-axis

Key Concepts

Shell Method

Washer Method

Setting up the Region and Limits of Integration

Watch next

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis