Find the area between the curves x = y^3 and x = y^2

Question

jennychan12 · Answer

is it just 
$$\int\limits_{0}^{1} x^\frac{ 1 }{ 3 } - x^\frac{ 1 }{ 2 } dx$$ ?

anonymous · Answer

So first of all, where do they intersect?

jennychan12 · Answer

oh sorry the limits are from 0 to 1

anonymous · Answer

Set x equations equal to each other: y³ = y² From there, we can see that y = 0 and 1

Set integration: $\displaystyle \int_0^1 y^2 - y^3 \space dy$

anonymous · Answer

You know, it's just as valid, and much easier, to integrate with respect to $y$.

jennychan12 · Answer

ohh, ok

jennychan12 · Answer

-_- the area is 0??

anonymous · Answer

No. How did you get 0, though?

jennychan12 · Answer

oh wait whoops sorry. my mistake.

jennychan12 · Answer

it's 1/12

anonymous · Answer

attachment

jennychan12 · Answer

lol

jennychan12 · Answer

wait one question....
sorry,
why is it y^2 - y^3 ?
i thought it was top curve - bottom curve..?

anonymous · Answer

Because in between y = 0 and y =1, y² is larger than y³
Let's saying that y = 1/2.
y² = 1/4
y³ = 1/8 

We can see that y² is larger than y³ in this interval so y² is top curve.

jennychan12 · Answer

oh ok, thanks again :)

anonymous · Answer

No problem.