Double Integrals in Polar Coordinates

1. Evaluate  the integral \(\displaystyle{ \iint_R 4x \, dA} \) over the region region \( R=\{(r,\theta): 1 \leq r \leq 2, 0 \leq \theta \leq \pi/2\}\).  solution

2. Evaluate  \(\displaystyle{ \iint_R  2xy \, dA} \), where \( R \) is the portion of the annular region \(4 \leq x^2+y^2\leq 25\) that lies in the first quadrant.   solution

3. Evaluate  \(\displaystyle{ \int_2^3 \int_0^x  \frac{x}{\sqrt{x^2+y^2}} \, dy\, dx} \)   solution

4. Evaluate  \(\displaystyle{ \int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}  (x^2+y^2)^2 \, dx\, dy} \)    solution

5. Find the volume of the solid bounded by the paraboloid \(z = 16-3x^2-3y^2\) and the plane \( z = 4\).  solution

6. Find the area enclosed by one loop of the four-leaved rose \( r = \cos(2\theta) \).    solution