Question

\(\int\limits_{ - 2}^2 {\int\limits_0^{\sqrt {4 - {y^2}} } {f(x,y)dy} dx} \)

Short answer

We should know how to do graphical representation.

Step-by-step solution

Given data:

Consider, \(\int\limits_{ - 2}^2 {\int\limits_0^{\sqrt {4 - {y^2}} } {f(x,y)dx} dy} \).

The inner limits of integration indicate that \(0 \le x \le \sqrt {4 - {y^2}} \)and the outer limits of integration indicate that \( - 2 \le y \le 2\). Note that the limits on \(y\)are the intersection points for the limits on \(x\). Typically in cases like this closed region is determined by the –boundary curves.

Graph the curves \(x = 0\) and \(x = \sqrt {4 - {y^2}} \).

Graph:

Original integral:

Now solve the equation \(x = \sqrt {4 - {y^2}} \)for \(y\) to get \(y = \pm \sqrt {4 - {y^2}} \). Use a vertical line intersecting the region of integration to help determine the \(y\)boundaries:

\( - \sqrt {4 - {x^2}} \le y \le \sqrt {4 - {x^2}} \)

The extreme values of are \(x\) 0 and 1.