March 09, 2007

Scribe Post

Finding the Volume of a Solid
(note: The pics aren't as clear, so please just click on them for a better view)

Imagine a rectangular, three-dimensional solid. To take the volume of this figure, we multiply its thickness, height and width together.

V = (thickness)(length)(width)

Another way to find V is if we divide the solid into even figures, find the volume of one of the smaller pieces, and multiply it by how many smaller pieces there are. In other words:

Volume has three dimensions, and Area has two. By integrating, we are essentially finding the thickness of an object. Going back to the equation V = (thickness)(length)(width), one dimension is already accounted for. We only need to find two more, and this is the area of the cross section.

In general, any function rotated around the x-axis will have circular cross sections, and its volume can be found by:

A(x) is the function representing the changing areas. The area for all the cross sections is represented by A = πr2, but r varies, so we need a function that represents r.

Let’s take a look at an example:

Imagine a line segment from the point (0, 4) to (8, 0). Revolve this line on the x-axis, and you’d get a cone, as seen in the diagram below. To find the volume, let’s follow the steps we learned in class:

(1) Find the function that represents the radius.
(2) Replace “r” with this function in the “volume equation” up there.
(3) Integrate over the interval.

If we compare this to what we get if we use the geometric equation we’ve used in elementary (V = 1/3πr2h), we get the same answer!!

So there you go folks. I think the three steps are doable. Homework is Exercise 8.2 ODDS. Scribe is JANN!

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