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 question, if you could detail the process and how to solve the integral

Answer 1

i)

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

= x^3/3-x^4/4

evaluated from x=0 to x=1

= 1/3-1/4 - 0 -0

= 4/12-3/12

= 1/12

the two curves intersect at x=0 and x=1

where

x^3=x^2

x^3-x^2 = 0

x^2(x-1) = 0

x=0, 1, and -1 (ignore -1 for quadrant I)

for quadrant I, the area between the curves = 1/12

ii)

y=x^3

solve for x

x=y^(1/3)

y=x^2

x=y^(1/2)

area = integral of [y^(1/3)-y^(1/2)]dy

= y^(4/3)/4/3- y^(3/2)/3/2

evaluated from y=0 to 1

=3/4-2/3- 0 -0

=9/12-8/12

= 1/12