Finding Volume of Solid w/ Semicircular Cross-Sections

[IntegrateMe]

x 0 0.5 1.0 1.5 2.0 2.5 3.0

f(x) 2 1.3 0.9 0.6 0.7 1.1 1.9

Find a formula for the volume V of the solid whose base is the region bounded by y = f(x), the x-axis, and the line x = 3 and its cross-sections perpendicular to the x-axis are semicircles.**

So, I plotted the points and got a graph that looks something like this:

http://i.imgur.com/AiFo6.jpg

Now to start on actually solving the problem.

So I figure that we should break the region up into a small dx pieces, and just sum up all of these pieces using an integral.

However, I'm having trouble figuring our what the area of each piece will be. Any help?

 

[LCKurtz]

x 0 0.5 1.0 1.5 2.0 2.5 3.0

f(x) 2 1.3 0.9 0.6 0.7 1.1 1.9

Find a formula for the volume V of the solid whose base is the region bounded by y = f(x), the x-axis, and the line x = 3 and its cross-sections perpendicular to the x-axis are semicircles.**

So, I plotted the points and got a graph that looks something like this:

http://i.imgur.com/AiFo6.jpg

Now to start on actually solving the problem.

So I figure that we should break the region up into a small dx pieces, and just sum up all of these pieces using an integral.

However, I'm having trouble figuring our what the area of each piece will be. Any help?

I'm guessing you don't understand exactly what the solid looks like. The region you have drawn is the base of the solid. Think of it as the floor. The solid itself stands on that base and has semicircular cross sections. Can you figure out the area of the cross sections at the given points? How do you calculate an integral with only finitely many points?

 

[Mark44]

x 0 0.5 1.0 1.5 2.0 2.5 3.0

f(x) 2 1.3 0.9 0.6 0.7 1.1 1.9

Find a formula for the volume V of the solid whose base is the region bounded by y = f(x), the x-axis, and the line x = 3 and its cross-sections perpendicular to the x-axis are semicircles.**

So, I plotted the points and got a graph that looks something like this:

http://i.imgur.com/AiFo6.jpg

Now to start on actually solving the problem.

So I figure that we should break the region up into a small dx pieces, and just sum up all of these pieces using an integral.

Yes and no. Yes to break up the region into small rectangular pieces of width Δx, and no to summing up the pieces using an integral. You don't have a formula for your function, so you aren't going to be able to find an antiderivative.

However, I'm having trouble figuring our what the area of each piece will be. Any help?

Are you asking about area or volume? You're going to have to add up the volumes of few semicircular slices. The volume of each of these slices will be the area of one semicircular face, times Δx.

The best approximation for the base rectangles will probably be to use the midpoint for each subinterval. You can interpolate a function value by averaging the function values at each end of that subinterval. That's what I would do.

 

[LCKurtz]

The best approximation for the base rectangles will probably be to use the midpoint for each subinterval. You can interpolate a function value by averaging the function values at each end of that subinterval. That's what I would do.

Or perhaps Simpson's rule?