Question

Using integration, find the value of `m`. If the area bounded by parabola `y^2=16ax` and the line `y=mx` is `a^2/12` square units.

Answer 1

`y^2=16an`
`y=mx`
`(mx)^2=16an`
`x=0,x=(16a)/m^2`
`int_0^(16/m^2)(sqrt(16ax)-mx)dx`
`sqrt(16a)int_0^(16/m^2)x^(1/2)dx-m int_0^((16a)/m^2)*xdx`
`sqrt(16a)*2/3*(8*9sqrta)/m^2-m/2*(256a^2)/m^4`
`64/3*a^2/m^3-(128a^2)/m^3`
`a^2/m^3{(8*64-3*128)/3}`
`a^2/m^3*128/3=a^2/12`
`m^3=(12*128)/3=4*128`
`m=8`.