Question

Using calculus, how would you find the equation for the volume of a pyramid that is not a right pyramid (the length and width are different sizes)? When the length and width are the same, it seems pretty easy, but when they're different, you have 3 variables, which I haven't really dealt with much.

Answer 1

Do you really have three different variables that you have to integrate though? Consider instead of a solid revolved around the x-axis instead make it a rectangle

Answer 2

Don't need calculus for this.
Volume = (1/3)(base area)*height
base area = L*W, therefore:

Volume = (1/3)(length*width)*height
http://www.mathsteacher.com.au/year10/ch14_measurement/25_pyramid/21pyramid.htm

Answer 3

@Wired did you read the first two words of his query?

Answer 4

Yes, which is why I stated that you don't need calculus. @alexray19, are you asking just a general question (in otherwords do you think you need to use calculus for this), or is what you posted the question asked of you (in otherwords, how would you solve this using calculus)?

Answer 5

Yes, I'm specifically wanting to know how to get that formula through calculus, and when the width and length are of different sizes. Or another way you could put it is: prove that pyramids with different widths/heights have the same volume formula as a pyramid with identical width/height.

Answer 6

@alexray19, Ok, that makes more sense.

Answer 7

Kainui: the problem with your response is that when you revolve a function around the x axis, at any give x value, the revolved part is the same distance all the way around. In the 2nd image you put, the distance of the surface from the x axis (as you go around the x axis) is varying, which adds another variable.

Answer 8

Thanks for the consideration though

Answer 9

@alexray19 Not quite. You are really only applying the area of a circle formula and integrating along a function. The difference is very trivial as you can just replace the circle with the area of a rectangle and place them outside as constants since they are not functions of height, as width and depth are separate.