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4. [20 pts] Let R be the region bounded by the graph of the function $f(x) = x^2$ and the curves $y = 0$ and $x = 2$. y R x 2 Let S be the three-dimensional solid obtained by revolving the region R around the x-axis. In this problem, we use the Washer Method to approximate the volume of the solid S using a Riemann Sum. A. Divide the x-interval [0, 2] into four equal subintervals. Label the x-values of the endpoints of each interval. On each subinterval, form a rectangle with height given by the value of $f(x)$ evaluated at the left endpoint of that subinterval. Draw these rectangles on the graph above. Make sure your drawings and labelings are done clearly and precisely. B. Calculate the height of each of the four rectangles. Give exact answers for each rectangle from left to right. height of rectangle 1 = height of rectangle 2 = height of rectangle 3 = height of rectangle 4 = C. Now rotate each of these rectangles about the x-axis to obtain a thickened washer. Find the volume of each washer. Give exact answers!

          4. [20 pts] Let R be the region bounded by the graph of the function $f(x) = x^2$ and the curves $y = 0$ and $x = 2$.
y
R
x
2
Let S be the three-dimensional solid obtained by revolving the region R around the x-axis. In this problem, we use
the Washer Method to approximate the volume of the solid S using a Riemann Sum.
A. Divide the x-interval [0, 2] into four equal subintervals. Label the x-values of the endpoints of each interval. On
each subinterval, form a rectangle with height given by the value of $f(x)$ evaluated at the left endpoint of that
subinterval. Draw these rectangles on the graph above. Make sure your drawings and labelings are done clearly
and precisely.
B. Calculate the height of each of the four rectangles. Give exact answers for each rectangle from left to right.
height of rectangle 1 = 
height of rectangle 2 = 
height of rectangle 3 = 
height of rectangle 4 = 
C. Now rotate each of these rectangles about the x-axis to obtain a thickened washer. Find the volume of each
washer. Give exact answers!

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4. [20 pts] Let R be the region bounded by the graph of the function f(x) = x^2 and the curves y = 0 and x = 2.
y
R
x
2
Let S be the three-dimensional solid obtained by revolving the region R around the x-axis. In this problem, we use
the Washer Method to approximate the volume of the solid S using a Riemann Sum.
A. Divide the x-interval [0, 2] into four equal subintervals. Label the x-values of the endpoints of each interval. On
each subinterval, form a rectangle with height given by the value of f(x) evaluated at the left endpoint of that
subinterval. Draw these rectangles on the graph above. Make sure your drawings and labelings are done clearly
and precisely.
B. Calculate the height of each of the four rectangles. Give exact answers for each rectangle from left to right.
height of rectangle 1 =
height of rectangle 2 =
height of rectangle 3 =
height of rectangle 4 =
C. Now rotate each of these rectangles about the x-axis to obtain a thickened washer. Find the volume of each
washer. Give exact answers!

Added by James S.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Adi S

09/14/2023

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4. [20 pts] Let R be the region bounded by the graph of the function f(x) = x² and the curves y = 0 and x = 2. y R X 2 Let S be the three-dimensional solid obtained by revolving the region R around the x-axis. In this problem, we use the Washer Method to approximate the volume of the solid S using a Riemann Sum. A. Divide the x-interval [0, 2] into four equal subintervals. Label the x-values of the endpoints of each interval. On each subinterval, form a rectangle with height given by the value of f(x) evaluated at the left endpoint of that subinterval. Draw these rectangles on the graph above. Make sure your drawings and labelings are done clearly and precisely. B. Calculate the height of each of the four rectangles. Give exact answers for each rectangle from left to right. height of rectangle 1 = height of rectangle 2 = height of rectangle 4 = height of rectangle 3 = C. Now rotate each of these rectangles about the x-axis to obtain a thickened washer. Find the volume of each washer. Give exact answers!

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Transcript

00:04 In this question, we are given the curve bounded between y is equal to 4 minus x square, x is equal to 2 and the line y is equal to 4 and y axis and we have to draw the graph of the reason bounded between these curves and find out the volume for them.

00:26 So let us first draw the graph.

00:29 So let us take this as y axis and let us take this as x axis or this is y axis, this is x axis.

00:43 So this is the point 0, 0 and let us draw y is equal to 4 minus x square.

00:52 So y is equal to 4 minus x square will be the parabola.

00:55 So let us say this is the point 0, 4 something like this.

01:03 So this is the point 0, 4 and now let us draw the line y is equal to 4.

01:10 So this will be the line y is equal to 4.

01:17 This is the line y is equal to 4 and this is our y axis.

01:22 Alright, so now the line x is equal to 2.

01:29 So this is the point 2, 0 and the line x is equal to 2 is given by this line.

01:43 This is the line x is equal to 2 and this is the reason bounded between all these curves and y axis.

01:54 Alright, this is the reason.

01:56 So let me just correct it.

02:05 This was y axis and this is how we have to rotate.

02:14 This reason about y axis and now we have to find out the volume.

02:18 Alright, so using the voscher's method, we use the voscher's method to find out the volume.

02:25 So using voscher's method.

02:29 So what does voscher's method says? so for voscher's method, by voscher's method, our volume is given by integration from a to b, a of y into dy.

02:45 Alright, so which is equal to integration a to b.

02:52 This is equal to pi r square, area is pi r square of any circle.

02:57 So this is pi into r square and the radius will be axis of rotation minus outer region, outer axis square minus pi into axis of rotation minus inner axis square into dx.

03:23 This is voscher's method.

03:24 So for using this method, we will find out the axis of rotation, the outer curve and the inner curve.

03:32 So axis of rotation, that is aor is equal to x is equal to 0, which is y axis.

03:47 Y axis is given to be axis of rotation in the question.

03:50 So the outer curve is by the graph, our outer curve...

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