Calculating Volume by Cross-Sections: A Riemann Sums Approach

[reconmaster]

Does anybody know how to start to workout this problem? The region of the base of the solid is bounded by y=abs(x)+3 and y=x^2-9, with parameters x=0 and x=4. Each cross-section is a square with its diagonal on the base. It also asked something about using Rieman Sums to solve this problem. I made the solid using foams and it looked like a hoof-shaped base embeded in squares. Thanks for any help.

 

[jdavel]

reconmaster,

Give us the original wording of the problem. Yours is too vague.

 

[arildno]

1.Are you sure your limits for x is right (0<=x<=4), or are you looking only at half the figure (-4<=x<=4)?

2. In order to solve this problem, you should find the centerline s(x) first.
Then, the infinitesemal volumelement is A(x)ds, where ds is the linelength at x,
ds=sqrt(1+s'(x)^(2))dx, whereas A(x) is the cross-sectional area.

 

[reconmaster]

The problem asks to find the volume by cross-sections, and simply gave those above equations. The limits are from -4 to 4 but the final volume was asked only of x=0 and x=4. I just assumed to solve this problem you find the volume of a whole bunch of slices and add them together. Is this approach correct?

 

[arildno]

That's right!
Note that the cross-sectional area A(x) is a square.
(I have integrated along the curved centerline s(x))

However, by "slicing" your volume differently, you can find the volume more easily.

About the Riemann sums, this is about using infinitesemal boxes stretching up in the vertical, for example.
Hence, we are not, in that case, using a cross-section method to compute the volume.

 

[mathwonk]

the basic principle of volumes by cross sections is that the volume is the integral of the area function for a slice.
You are given that all the slices are squares and are given the diameter of every square. So just compute the area of a square from its diagonal, write that as a function of x, and integrate that function of x from x = 0 to x=4.