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

arishorts

1. The problem statement, all variables and given/known data


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


2. Relevant 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

3. 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

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

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?