Question

If * (x, y) = 30(y + y ^ 2) represents the population density of a planar region on Earth, where randy are measured in miles, find the number of people in the region bounded by the curves x = y ^ 2, x = 2y - y ^ 2

Answer 1

When do the curves $x=y^2$ and $x=2y-y^2$ intersect?

$$y^2=2y-y^2$$

$$2y(y-1)=0$$

$$y_1=0, y_2=1$$

Express the number of people in the region as iterated integral

$$\displaystyle\int_{0}^{1}\displaystyle\int_{y^2}^{2y-y^2}30(y+y^2)dxdy$$

$$=\displaystyle\int_{0}^{1}30(y+y^2)[x]\begin{matrix} 2y-y^2 \\ y^2 \end{matrix}dy$$

$$=\displaystyle\int_{0}^{1}30(y+y^2)(2y-2y^2)dy$$

$$=\displaystyle\int_{0}^{1}60(y^2-y^4)dy$$

$$=[20y^3-12y^5]\begin{matrix} 1\\ 0 \end{matrix}$$

$$=20-12=8$$

The number of people in the region bounded by the curves is $8.$