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.
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` .