Evaluate the given integral by changing to polar coordinates. ∫ ∫ sin(x2 + y2) dA
R

where R is the region in the first quadrant between the circles with center the origin and radii 1 and 2

The result of the integral is

Step-by-step explanation:

Polar coordinates:

In polar coordinates, we have that:

In which r is related to the radius values, while is related to the angles in the trigonometric circle.

In this question:

R is the region in the first quadrant

In the first quadrant in the trigonometric circles, the angles go from 0 to , which means that this are the outer limits of integration.

Between the circles with center the origin and radii 1 and 2

This means that the inner limits of integration are between 1 and 2.

The integral will be given by:

Inner integral:

By substituion,

So

= \frac{1}{2} \int_{1}^{2} \sin{u} du[/tex]

Integral of sine is minus cosine. So

Before replacing, we substitute back u.

Outer integral:

The result of the integral is

answer: the answer is 127 degrees