Find the volume of a pyramid with height 23 and rectangular base with dimensions 2 and 10.  Find the volume using integrals. Do it the way that it would be done in calculus 2.

Expert Answers
sciencesolve eNotes educator| Certified Educator

You should sketch a pyramid, centered on y axis, whose top is located at origin.

You should write the area of a cross section, as a function of y, hence, using similar triangles yields the following reports such that:

`l/L = w/W = y/h`

l represents the length of cross section

L represents the length of the base

w represents the width of cross section

W represents the width of base

h represents the height of pyramid

Considering the ratio `l/L = y/h ` yields:

`l = (L/h)y = (10/23)y`

Considering the ratio `w/W = y/h`  yields:

`w = (W/h)y = (2/23)y`

You should evaluate the area of cross section such that:

`A(y) = (10/23)y*(2/23)y => A(y) = 20/529 y^2`

You should use the following formula to find the volume of pyramid such that:

`V = int_0^h 20/529 y^2 dy => V = (20/529)int_0^23 y^2 dy`

`V = (20/529)*y^3/3|_0^23 => V = (20/(23^2*3))(23^3)`

`V = (20*23)/3 => V = 460/3`

Hence, evaluating the volume of pyramid using integral yields  `V = 460/3` .