When I try Find the volume of the region $R$ lying below the plane $z = 3-2y$ and above the paraboloid $z = x^2 + y^2$
Solving the 2 equations together yields the cylinder $x^2 + (y+1)^2 = 4 $ How do I get the volume then???
When I try Find the volume of the region $R$ lying below the plane $z = 3-2y$ and above the paraboloid $z = x^2 + y^2$
Solving the 2 equations together yields the cylinder $x^2 + (y+1)^2 = 4 $ How do I get the volume then???
First of all, I draw a plot for $x^2+(y+1)^2=4$ or $r^2+2r\sin(\theta)=3$ which is our integration area on plane $z=0$.
You see that $r$ varies from $r=3$ to $r=-\sin(\theta)+\sqrt{\sin(\theta)^2+3}$ and $\theta$ from $0$ to $pi/2$. As the volume is symmetric so you should double the result.