Mathematics

Problem

A straight line through the origin with positive slope

m

meets the parabola

y=x^{2}

at two points, one of which is the origin and the other

(m,m^{2})

. The region

P

is enclosed by the line

y=mx

, the parabola

y=x^{2}

, and the

x

-axis between

x=0

and

x=m

. If the area of

P

equals

\dfrac{4}{3}

square units, a. (i) Obtain a single definite integral that represents the area of

P

in terms of

m

. b. (ii) Hence show that

m^{3}=8

and determine the value of

m