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

1. RolyPoly

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 $\int_{-1}^{0}(x^2-x^4)dx + \int_0^1(x^2-x^4)dx$

2. RolyPoly

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$

3. anneflys

Wait, so why did you multiply everything by 2?

4. anneflys

?

5. RolyPoly

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.