0
$\begingroup$

While solving a simple problem for finding are of the region bounded by $x=y^2$ and $x=y$. Are the following correct limits?


When $x$ is the outer integration variable $$\int^1_0\int^\sqrt{x}_xdydx$$

When $y$ is the outer integration variable?

$$\int^1_0\int^y_{y^2}dxdy$$

The reason for asking the question is, I am getting the different answer for both the cases which should not be the case.

$\endgroup$
3
$\begingroup$

\begin{align} \int_0^1 \int_x^{\sqrt{x}} \, \, dydx &= \int_0^1 x^\frac12-x \, dx \\ &= \frac23 -\frac12 \\ &= \frac16 \end{align}

\begin{align} \int_0^1 \int_{y^2}^{y} \, \, dxdy &= \int_0^1 y-y^2 \, dx \\ &= \frac12 -\frac13 \\ &= \frac16 \end{align}

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.