Question

Compute ∬D(x+y)dA where D={(x,y)∈R^2:0≤x≤2 and 0≤y≤√(4-x^2)}


.

Select one:

a. 16

b. 8

c. 8/3

d. 16/3

e. 3

Answer 1

Using Fubini's theorem, we get:

$\iint_D(x+y)dA=\int_0^2dx\int_{0}^{\sqrt{4-x^2}}(x+y)dy=\int_0^2((xy+\frac{y^2}{2})\biggr\rvert_{0}^{\sqrt{4-x^2}})dx=$

$=\int_0^2(x\sqrt{4-x^2}+\frac12(4-x^2)))dx=-\frac12\int_0^2\sqrt{4-x^2}d(4-x^2)+\int_0^2(2-\frac12x^2)dx=$

$=\frac83+4-\frac43=\frac{16}3$

Answer:d). $\frac{16}3$