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

[lollikey]

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

 

[lollikey]

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

 

[SteamKing]

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(x_n) + 2Σ f(x_2j) + 4Σ f(x_2j-1)]

The Attempt at a Solution

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