Question

Let D={(x,y)∈R^2:y^2<x<y and 0<y<1}. The answer to ∬D 2y^2xdA


is

Select one:

a. 70

b. 1/35

c. -70

d. 2/35

e. 35

Answer 1

Since region bounded by

$x:y^2\to y,\ \ \ \ \ \ \ \ y :0\to 1$

Then

$\begin{aligned} \int\int_{D} y^2xdA&= \int_{0}^{1}\int_{y^2}^{y} y^2xdxdy \\ &= \frac{1}{2}\int_{0}^{1} y^2x^2\bigg|_{y^2}^{y} dy \\ &=\frac{1}{2}\int_{0}^{1} y^2(y^2-y^4)dy\\ &= \frac{1}{2}\int_{0}^{1} (y^4-y^6)dy\\ &= \frac{1}{2} (\frac{1}{5}y^5-\frac{1}{7}y^7)\bigg|_{0}^{1}\\ &= \frac{1}{2}( \frac{1}{5} -\frac{1}{7})\\ &= \frac{1}{35} \end{aligned}$