Reverse Order of Integration Question Reverse the order of integration of the following integral:
$\int_{0}^{2}$ $\int_{2x}^{4x-x^2}$$dydx$
I'm not quite sure what to do about the term $4x - x^2.$ My understanding for solving these is that they typically require an inversion of the term, however, while it's easy to express $y = 2x$ as $x = \frac{1}{2}y$, it's not possible to define $y = 4x - x^2$ in terms of $x$.What is the key to solving a question like this?


This is the graph of what your double integral is showing. Since this is a double integral of 1, this is computing the area of the arc-shape made. When the order is $\mathrm dy\,\mathrm dx$, the limits of integration for $y$ uses variables while the limits of integration for $x$ use constants. Since you want to change it to $\mathrm dx\,\mathrm dy$, it will be the opposite.
For $\mathrm dx\,\mathrm dy$ integration, span the curve from the left to the right by solving each equation for $x$. The limits of integration for $x$ will be $0$ to $4$.
$$\int_0^4\int_{2-\sqrt{4-y}}^{\frac12y}1\,\mathrm dx\,\mathrm dy$$
