Simpson's Rule to find the volume of f(x) rotated about the x and y axis.

[arishorts]

Homework Statement

Answers in the back of the book
about x-axis= 190
about y-axis= 828

Homework Equations

Simpson's Rule: (dx/3)* sum of(sequence of coefficients {1,4,2...2,4,1}*sequence of function values{f(0), f(1), f(2)...f(n-2),f(n-1), f(n)})

Volume using Shells: 2π ∫ (radius)(height) dx

Volume using Cross-Sections: π ∫ (outer radius)^2 - (inner radius)^2 dx

The Attempt at a Solution

I found the area (≈) under the curve using Simpson's law, how do i rotate it without a given f(x)? The book doesn't ask for the specific method (either shells or cross-sections), but I'd like to understand how to do both. Please help.

 

[LCKurtz]

I found the area (≈) under the curve using Simpson's law, how do i rotate it without a given f(x)? The book doesn't ask for the specific method (either shells or cross-sections), but I'd like to understand how to do both. Please help.

The integral for a volume of revolution of $f(x)$ between $a$ and $b$ is$$
V=\pi\int_a^b f^2(x)dx$$Do Simpson's rule on that, not on the area integral.

 

[arishorts]

we're not given f(x) though. We're only given the values of f(x)

 

[LCKurtz]

we're not given f(x) though. We're only given the values of f(x)

So you can figure out the values of $\pi f^2(x)$ and do Simpsons rule just like you did for the area.

 

[z37002]

Great question. Would someone please give more details as to combining function and s rule?