Answer: The transformed integral is
The region of integration is a triangle delimited by the 
axis, the vertical line 
, and the line 
.
Step-by-step explanation: For this integral we have:
1. 
and
2. 
.
The first condition says that the region of integrarion is within the "slice" of the xy plane between the polar angle of 
and 
in the first quadrant. To make sense of the second condition we will convert to the Cartesian coordinates. Remember that
3. 
4. 
5. 
6. 
Using 2. (squaring it), along with 5. we obtain
Now using 3. to express 
we obtain
The left inequality 
just says that we start from the origin. The right inequality, after cancelling 
, yields 
. Since we are in the first quadrant has to be positive so this inequality can be rewritten as (taking the square root) 
. So the region is a triangle delimited by the slice of the first quadrant between and and by the vertical line (see figure).
In other words, this triangle is delimited by the x axis, by the vertical line x=2 and by the line 
This helps us to define the boundaries of integration: we have 
(from the origin up to x=2) and 
(from the x axis i.e. y=0 up to the line mentioned before).
Next we transform the area element. In cartesian coordinates it is 
while in polar coordinates it is 
so we have 
Next we use 4. to express the sine: 
.
Finally, we use 5. 
.
Therefore, the transformed integral is
