Question

Suppose that ( f(x, y)=frac{y}{1+x} ) at which ( {(x, y) mid 0 \leq x leq 3,-x leq y leq sqrt{x}} ). The plot of the region is below. Then the double integral of ( f(x, y) ) over ( D )

Answer 1

To find the double integral of the given function over a given region, the bounds for x and y are determined from the region. The double integral is then written using these bounds.

Step-by-step explanation:

The given function is f(x, y) = (y)/(1+x) and the region D is defined as (x, y) .

To find the double integral of f(x, y) over D, we need to set up the integral using the bounds of the region.

1. The lower bound for x is 0.
2. The upper bound for x is 3.
3. The lower bound for y is -x.
4. The upper bound for y is sqrt(x).

Using these bounds, the double integral of f(x, y) over D can be written as:

∫∫(0 ≤ x ≤ 3)(∫(-x ≤ y ≤ sqrt(x))((y)/(1+x)) dy) dx