What is the area between the graphs?

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Answer 1 · 2018-05-23T22:02:34

#Ω=5/12m^2#

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#Ω=int_0^1(root(3)(x)-x^2)dx=#

#int_0^1root(3)(x)dx-int_0^1x^2dx=#

#int_0^1x^(1/3)dx-int_0^1x^2dx=#

#[3/4x^(4/3)]_0^1-[x^3/3]_0^1#

#3/4-1/3=5/12m^2#

Answer 2 · 2018-05-23T22:09:37

#5/12#

the integral is the area between the blue and red curves from the green line #(x=0)# and the orange line #(x=1)#:

the area is #int_0^1(x^(1/3)-x^2)dx# (subtract #x^2# from #x^(1/3)# because #x^(1/3)# is always greater on #0<x<1#)

solving:
#int_0^1(x^(1/3)-x^2)dx=F(1)-F(0)#, where #F(x)=3/4x^(4/3)-1/3x^3#

#=3/4(1)^(4/3)-1/3(1)^3-3/4(0)^(4/3)+1/3(0)^3#
#=3/4-1/3=5/12#