Double integral in polar coordinates

Determine the intervals for  $r$   and  $\theta$  associated with the blue region.

•  $3\le r\le4,\ 0\le\theta\le\pi$  

•  $2\le r\le4,\ 0\le\theta\le\frac{\pi}{4}$  

•  $\pi\le r\le\frac{\pi}{4},\ 1\le\theta\le4$  

•  $2\le r\le4,\ \frac{\pi}{4}\le\theta\le\pi$  

Formula for the double integral in polar coordinates is

• $\int_{ }^{ }\int_{ }^{ }f\left(\left(r,\theta\right)\right)r\ drd\theta$

• $\int_{ }^{ }\int_{ }^{ }f\left(x,y\right)\ dA$

• $\int_{ }^{ }\int_{ }^{ }f\left(\left(r,\theta\right)\right)d\theta\ dr$

• $\int_{ }^{ }\int_{ }^{ }f\left(\left(r,\theta\right)\right)\theta\ drd\theta$

Convert  $\int_0^{\frac{1}{\sqrt{2}}}\int_x^{\sqrt{1-x^2}}\left(3x\ +4y^2\right)\ dydx$  into polar coordinates

•  $\int_0^{\frac{\pi}{4}}\int_0^13r+4r\sin\theta\ drd\theta$  

•  $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\int_0^1r^2\left(3\cos\theta+4r\sin^2\theta\right)drd\theta$  

•  $\int_0^{\frac{\pi}{2}}\int_0^13r\cos\theta+4r^2\sin^2\theta\ drd\theta$  

•  $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\int_0^13\cos\theta+4r^2\sin^2\theta\ r\ drd\theta$  

R is the region in the upper half-plane bounded by the circles

 $x^2+y^2=1$   and  $x^2+y^2=4$  . Find the area of R

•  $\frac{\pi}{2}$  

•  $\frac{3\pi}{2}$  

•  $2\pi$  

•  $\frac{5\pi}{2}$  

Find the volume of the solid bounded by the plane z = 0 and the paraboloid $z=1-x^2-y^2$ 

•  $\frac{\pi}{2}$  

•  $\frac{\pi}{5}$  

•  $\frac{\pi}{3}$  

•  $\frac{\pi}{4}$