Answer :
Given :
Given a curve , [tex]y=2x^2[/tex] .
To Find :
The area of the region between the given curve and the x-axis on the interval [0, b] .
Solution :
Now , area under the curve is given by :
[tex]A=\int\limits^b_0 {2x^2} \, dx \\\\A= |_0^b(\dfrac{2}{3}x^{(2+1)})\\\\A=\dfrac{2b^3}{3}[/tex]
( Integration of [tex]x^2[/tex] is [tex]\dfrac{x^3}{3}[/tex] )
Therefore , the region between the given curve and the x-axis on the interval [0, b] is [tex]\dfrac{2b^3}{3}[/tex] .
Hence , this is the required solution .