Question

Find the area of the region bounded by the curves y=x^2 and y=x^4

Answer 1

1. find the intercept of the curve
\[x^4 - x^2 =0\]\[x^2 (x^2-1)=0\]\[x=0, 1, -1\]
Then integrate it
\[\int_{-1}^{0}(x^2-x^4)dx + \int_0^1(x^2-x^4)dx\]

Answer 2

Or...
since it is an even function,
\[\int_{-1}^{0}(x^2-x^4)dx + \int_0^1(x^2-x^4)dx\]\[=2\int_{-1}^{0}(x^2-x^4)dx\]

Answer 3

Wait, so why did you multiply everything by 2?

Answer 4

?

Answer 5

I didn't multiply everything by two...
It's just \[\int_{-1}^{0}(x^2-x^4)dx = \int_0^1(x^2-x^4)dx\] (Since it is an even function)
So,
\[\int_{-1}^{0}(x^2-x^4)dx + \int_0^1(x^2-x^4)dx\]\[=\int_{-1}^{0}(x^2-x^4)dx +\int_{-1}^{0}(x^2-x^4)dx \]\[=2\int_{-1}^{0}(x^2-x^4)dx \]Using the property of even function can save some work.