Question

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

Answer 1

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

Answer 2

So first of all, where do they intersect?

Answer 3

oh sorry the limits are from 0 to 1

Answer 4

Set x equations equal to each other: y³ = y² From there, we can see that y = 0 and 1

Set integration: \(\displaystyle \int_0^1 y^2 - y^3 \space dy\)

Answer 5

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

Answer 6

ohh, ok

Answer 7

-_- the area is 0??

Answer 8

No. How did you get 0, though?

Answer 9

oh wait whoops sorry. my mistake.

Answer 10

it's 1/12

Answer 11

lol

Answer 12

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

Answer 13

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.

Answer 14

oh ok, thanks again :)

Answer 15

No problem.