Multiple Integrals; Coordinate Systems
Exercise 1 : Multiple Integrals with Constant Limits
Evaluate the following multiple integrals:
- $\int_{y=0}^{b} \int_{x=0}^{a} dx dy$
- $\int_{y=0}^{2} \int_{x=0}^{1} x^2 dx dy$
- $\int_{x=0}^{\pi} \int_{y=0}^{\pi} \sin x \sin y dx dy$
- $\int_{n=1}^{2} \int_{v=2}^{4} n(1+v) dv dn$
- $\int_{x=-1/2}^{1/2} \int_{y=-1}^{1} \int_{z=0}^{2} dx dy dz$
- $\int_{x=0}^{1} \int_{y=0}^{y_1} \int_{z=0}^{z_1} e^{az} dx dy dz$
Exercise 2 : Multiple Integrals with Variable Limits
Evaluate the integrals:
- $\int_{x=0}^{2} \int_{y=x-1}^{3x} x^2 dy dx$
- $\int_{x=0}^{1} \int_{y=0}^{2x} \int_{z=0}^{x+y} dx dy dz$
- Using a double integral, obtain the area of an ellipse and the position of the center of mass of the half ellipse $(x \geq 0)$. The equation of an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
Exercise 3 : Coordinate Systems
- A point has Cartesian coordinates $P = (3, 3)$. What are its polar coordinates?
- Give the equation of a circle of radius $R$ in Cartesian coordinates and polar coordinates.
- Obtain the equation of the spiral shown in Fig. 13.22 in polar coordinates.
- Evaluate $\int_{\theta=0}^{\pi/4} \int_{r=0}^{a} r^2 \cos \theta , dr , d\theta$.
Exercise 4 : Cylindrical Coordinates
- Compute the volume of the hollow cylinder shown in Fig. 13.23 using cylindrical coordinates.
- Evaluate the volume of a cone of radius $R$ and height $h$. Obtain the moment of inertia of the cone about its center axis. The density $\rho$ is a constant.
Exercise 5 : Moment of Inertia
Calculate the moment of inertia of a sphere of radius $R$ and of constant density $\rho$ about an axis through its center, using spherical coordinates.