How do you find the volume of a pyramid using integrals?

Question

Elijah Abram · Accepted Answer

Let us find the volume of a pyramid of height #h# with a #b	imes b# square base.  If #y# is the vertical distance from the top of the pyramid, then the square cross-sectional area #A(y)# can be expressed as #A(y)=(b/hy)^2=b^2/h^2y^2#. So, the volume #V# can be found by the integral #V=int_0^hA(y) dy=b^2/h^2int_0^hy^2 dy=b^2/h^2[y^3/3]_0^h
=1/3b^2h#. I hope that this was helpful.

Elijah Abram · Answer

undefined To find the volume $V$ of a pyramid using integrals, you can use calculus to derive the formula. The volume of a pyramid with a base area $A$ and height $h$ can be expressed as $\frac{1}{3}Ah$.

To derive this using integrals, you can consider the pyramid as a stack of infinitesimally thin slices parallel to its base. Each slice has an area that varies with its distance from the apex of the pyramid.

Integrate the area of each slice over the height of the pyramid, from 0 to $h$, using the variable $y$ to represent the distance from the apex to the slice. The area of each slice can be expressed in terms of $y$, and then integrated with respect to $y$.

The general formula for the area of a slice at height $y$ can be determined using similar triangles or other geometric methods, depending on the shape of the pyramid. Once you have the formula for the area of the slice in terms of $y$, integrate it with respect to $y$ from 0 to $h$.

The integral will give you the volume of the pyramid as the result.

So, in summary, to find the volume of a pyramid using integrals, express the area of each slice as a function of its height, then integrate these areas with respect to height over the entire height of the pyramid.