Volume of Solid Rotated about Y-Axis: Estimate w/Simpson's Rule

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In summary, the problem asks us to use Simpson's Rule with n = 8 to estimate the volume of the solid created by rotating the given figure about the y-axis. To do so, we must first calculate the area under the curve using Simpson's Rule, and then use the Second Theorem of Pappus to find the volume.
  • #1
lollikey
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Homework Statement


(a) If the region shown in the figure is rotated about the y-axis to form a solid, use Simpson's Rule with n = 8 to estimate the volume of the solid. (Round your answer to the nearest integer.)

Homework Equations


delta(x) = b-a/n
delta(x)/3 [ f(x) + 4f(x)+ 2f(x) + f(x)][/B]

The Attempt at a Solution

 

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  • #2
my attempt at the solution was

delta(x) = 10-2/8 = 1

1/3[ 1+2(1.5)+4(2)+2(2)+4(3)+2(3.5)+4(4)+2(3.5)+1] = 59/3
 
  • #3
lollikey said:

Homework Statement


(a) If the region shown in the figure is rotated about the y-axis to form a solid, use Simpson's Rule with n = 8 to estimate the volume of the solid. (Round your answer to the nearest integer.)

Homework Equations


delta(x) = b-a/n
delta(x)/3 [ f(x) + 4f(x)+ 2f(x) + f(x)]

The formula you have is incorrect for Simpson's First Rule, or at least, it is not written properly.

Let's stipulate that h = common interval = Δx = (b - a) / n, where a and b represent the x values of the start and finish, respectively, of the x-interval, and n is the number of intervals.

Then the area under the curve from x = a to x = b is

A = h * [f(x0) + f(xn) + 2Σ f(x2j) + 4Σ f(x2j-1)]

The Attempt at a Solution


lollikey said:
my attempt at the solution was

delta(x) = 10-2/8 = 1

1/3[ 1+2(1.5)+4(2)+2(2)+4(3)+2(3.5)+4(4)+2(3.5)+1] = 59/3

The first and last ordinates of the shaded area are both equal to 0, not 1.

Remember, the problem is asking to find the volume of the solid created by rotating the figure about the y-axis. Calculating the area under the curve is necessary, but not sufficient, to answer this problem.

To calculate the volume, you'll need to use the Second Theorem of Pappus in addition to Simpson's Rule:

http://mathworld.wolfram.com/PappussCentroidTheorem.html
 

Related to Volume of Solid Rotated about Y-Axis: Estimate w/Simpson's Rule

1. What is Simpson's Rule?

Simpson's Rule is a numerical method for estimating the area under a curve by dividing the curve into smaller sections and approximating each section as a parabolic curve.

2. How is Simpson's Rule used to estimate the volume of a solid rotated about the y-axis?

Simpson's Rule can be used to estimate the volume of a solid rotated about the y-axis by approximating the cross-sectional area of the solid using parabolic curves and then integrating these areas over the given interval.

3. Why is Simpson's Rule a more accurate method for estimating volume compared to other numerical methods?

Simpson's Rule is more accurate than other numerical methods because it takes into account the curvature of the curve being integrated, resulting in a better approximation of the actual volume.

4. Can Simpson's Rule be used to estimate the volume of any solid rotated about the y-axis?

Yes, Simpson's Rule can be used to estimate the volume of any solid rotated about the y-axis as long as the cross-sectional area can be approximated using parabolic curves.

5. Are there any limitations to using Simpson's Rule for estimating volume?

One limitation of using Simpson's Rule for estimating volume is that it can only be used for solids with rotational symmetry about the y-axis. It also requires a sufficient number of intervals to achieve an accurate approximation.

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