Tutorial 11

Tutorial 11: Multiple Integrals in Polar Coordinate & Its Engineering Application

Tutorial Question

1. In the following exercises, change the cartesian integral into an equivalent polar coordinate integral. Then solve the integral in polar coordinate:

   a. $\displaystyle\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} d y d x$

   b. $\displaystyle\int_{0}^{2} \int_{0}^{x} y d y d x$

   c. $\displaystyle\int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \frac{2}{\left(1+x^{2}+y^{2}\right)^{2}} d y d x$

   d. $\displaystyle\int_{0}^{\ln 2} \int_{0}^{\sqrt{(\ln 2)^{2}-y^{2}}} e \sqrt{x^{2}+y^{2}} d x d y$

2. Evaluate the $\iint 1-x^{2}-y^{2} d A$ using polar coordinates

3. Find the volume below $z=\frac{y^{2}}{x^{2}+y^{2}}$, above $x y$-plane and between cylinder $x^{2}+y^{2}=1$ and $x^{2}+y^{2}=2$

4. Find the volume between the sphere $x^{2}+y^{2}+z^{2}=1$ and the cone $z=\sqrt{x^{2}+y^{2}}$

5. Volume is equal to area only if the height (z) is equal to 1 . Find the area of ' $R^{\prime}$ where ' $R$ ' is the region bound by $r=3 \cos \theta$.

6. $\iiint y d V$, a solid is bound by $z=4-x^{2}-y^{2}$ in the first octant $(x=0, y=0, z=$ $0)$

7. Use cylindrical coordinates to find the volume of a curved wedge cut out from a cylinder $\left(x^{2}-2\right)^{2}+y^{2}=4$ by the planes $z=0 \mathrm{z}=0$ and $z=-y$.

   

8. Consider the region $E$ inside the right circular cylinder with equation $r=2 \sin \theta$, bounded below by the $r \theta$-plane and bounded above by the sphere with radius 4 centered at the origin. Set up a triple integral over this region with a function $f(r, \theta, z)$ in cylindrical coordinates.

   

9. Find the volume of solid bound by $z=2$ and $z=\sqrt{x^{2}+y^{2}}$

10. Use spherical coordinates to find the volume of the region outside the sphere $\rho=2$ $\cos (\phi)$ and inside the sphere $\rho=2$ with $\phi \in[0, \pi / 2]$.

    

11. Given a solid bound by $z=2$ and $z=\sqrt{x^{2}+y^{2}}$, find the mass density if the mass density is directly proportional to the square of the distance from origin.

12. Find the mass of ' $T$ ',$\rho(x, y, z)=y$, where $\mathrm{T}$ is region bound by $y=x^{2}+z^{2}$ and $y=4$.