Find the area of the region in quadrant I bounded between the curves y = x ^2 and y = x ^3 , using (i) Representative rectangles of width dx (ii) Representative rectangles of width dy

Question

Calculus problem, if you could detail the process and how to set up the integral.

Answer 1

Area = the same either way = 1/12

= integral of (x^2-x^3)dx

evaluated from x=0 to x=1

= x3/3 -x^4/4

=1/3-1/4 = 1/12

or

= integral of [(x^(1/3) -x^(1/4)]dy

evaluated from y=0 to y=1

= x^(5/4)/(5/4)-x^(4/3)/(4/3)

=4/5-3/4

= 1/12