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I know how to solve the integral but I"m a bit confused about what to sketch. Can someone help me out?

The integral I need to solve is this:

$$\int_0^1 \int_{x^{2}}^x (1 - 2xy)dydx$$

$$\int_0^1 \left[(y - xy^{2}) \right]_{x^{2}}^xdydx$$

$$\int_0^1 (x-x^{3}) - (x^{2} - x^{5}) dx$$

$$\int_0^1 x - x^{2} - x^{3} + x^{5} dx$$

$$ \left[( \frac{x^{2}}{2} - \frac{x^{3}}{3} - \frac{x^{4}}{4} + \frac{x^{6}}{6} \right]_0^1$$

$$ \frac{1}{2} - \frac{1}{3} - \frac{1}{4} + \frac{1}{6} = \frac{1}{12}$$

But how do I sketch this? How does the integrand play into the sketch?

Jwan622
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2 Answers2

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I think you can just plot the domain and the function:

enter image description here

Here of course $0 \leq x \leq 1$. For any value of $x$, $y$ obeys $x^2 \leq y \leq x$. This is the domain, drawn in blue on the plane $z=0$. The function itself is $z = 1 - 2 x y$, which is the orange surface that goes above as well as below $z=0$. The integral is the (vertical) volume between the blue and orange surfaces.

David G. Stork
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[enter image description here] I know image is not bright cause they only let you upload the image of 2mb maximum. Although i made it clear your sketch in image . [image now upgraded - Ed.]

Hanif Ashrafi
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