Math 121a homework 5 calculate the volume of the

Question

Math 121A: Homework 5-

Part 1: Multiple integration

1. (a) Calculate the volume of the three-dimensional object given by the points (x, y, z) that satisfy x² + y² ≤ 1 and x² + z² ≤ 1.

(b) Optional for the enthusiasts. Calculate the volume of the three-dimensional object given by x² + y² ≤ 1, x² + z² ≤ 1, y² + z² ≤ 1.

Part 2: Sturm-Liouville theory

2. Find the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville problem

d/dx[-e^-2xdy/dx] - e^-2x y = λe^-2xy

on the interval 0 ≤ x ≤ π, subject to the boundary conditions

y(0) = 0,  y'(π) - y(π) = 0.

Verify that the eigenfunctions satisfy the orthogonality relation, so that (y_m, y_n) = 0 for any m ≠ n.

3. The concentration of gas in a channel over the range 0 < x < a follows the diffusion equation

∂f/∂t = b ∂²f/∂x².

At x = 0 any gas is removed, and at x = a there are is a wall that is impermeable to gas, so suitable boundary conditions are

f(0, t) = 0,   ∂f/∂x (a, t) = 0.

(a) Solve the system by searching for separable solutions of the form f(x, t) = X(x)T(t). (This will result in a Sturm-Liouville problem for X.)

(b) Write down the solution f(x, t) to the equation for the initial condition

f(x, 0) = sin πx/2a + 1/5 sin 17πx/2a

For the case of a = π and b = 1, plot f(x, t) for the values of t of 0, 0.2, 0.5, 1.0, 2.0, and 5.0. Interpret the graphs physically.