## Midpoint Rule?

Math help on Calculating limits of functions, Differentiation of a product, Differentiation of a quotient, Composition of functions, Chain Rule, Variations, Extremums: Maximums, Minimums, Increasing Functions, Decreasing Functions, Constant Functions, Differential Equations, Growth of a Population, Formulas, Integration by Parts, Change of Variables; Double Integrals, Triple Integrals, Multiple Integrals, Integrals with a Parameter, Area, Volume, Approximations on My Math Forum.

### Midpoint Rule?

I got a homework problem that says to use the midpoint rule to approximate a volume of a solid, (finding an area of a given curve then rotated around the x-axis).

I looked up the 'midpoint rule' in my textbook and it only shows how to use it to approximate an area under a curve which I know how to do. But how do I find the volume of a function rotated around the x-axis using the Midpoint Rule.
A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.
Albert Einstein
jcmc2112
Queen of Hearts

Posts: 65
Joined: Fri Jul 31, 2009 10:15 pm
Location: New Jersey

### Re: Midpoint Rule?

When your text uses the midpoint rule to approximate the area under a curve it is (and they may even say so) constructing a rectangle and taking that area. For the case of a solid of revolution, the midpoint rule will give you a value to use as the radius of a cylinder. Just plug that into the formula for the volume of a cylinder. (pi r^2 h)
When I am working on a problem, I never think about beauty but when I have finished, if the solution is not beautiful, I know it is wrong. -- R. Buckminster Fuller
cmusick
King of Diamonds

Posts: 192
Joined: Thu Jan 15, 2009 6:27 am
Location: Japan

### Re: Midpoint Rule?

So its just like the Simpson's Rule that I learned in high school? Because I knew it was similar but didn't know exactly what it was because i have never heard of it .

A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.
Albert Einstein
jcmc2112
Queen of Hearts

Posts: 65
Joined: Fri Jul 31, 2009 10:15 pm
Location: New Jersey

### Re: Midpoint Rule?

Okay so I used the midpoint rule, simpsons rule, and the trapezoidal rule to estimate the area under this curve, which was 21, 20, and 20, respectively.

It is rotated around the x-axis, from the intervals: x=2 to x=10

(NOTE: This is not a regular function, just a plain old curve that the author came up with but the range is clearly visible to be able to use the above methods.)

How do I use this to estimate the volume?

I kept trying to use the [pi]r^2 h, but my answer seems to be ALOT larger than the answer provided in the textbook (which says it to be 169).
A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.
Albert Einstein
jcmc2112
Queen of Hearts

Posts: 65
Joined: Fri Jul 31, 2009 10:15 pm
Location: New Jersey

### Re: Midpoint Rule?

If the area is 21 (midpoint rule) and the width is 8 (from 10-2), then the 'radius' is 21/8, and the height is 8.

pi * (21/8)^2 * 8 is approximately 173. I'm assuming you did some rounding which would account for the difference from 169.
When I am working on a problem, I never think about beauty but when I have finished, if the solution is not beautiful, I know it is wrong. -- R. Buckminster Fuller
cmusick
King of Diamonds

Posts: 192
Joined: Thu Jan 15, 2009 6:27 am
Location: Japan