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 $F(x)=3/4x^(4/3)-1/3x^3$

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