Let's solve problems 3, 4, and 5 from the image.

Problem 3: Rectangular Plate

We are given a rectangular plate over the region: $\Omega = \{ (x, y) \mid 0 \leq x \leq 20, 0 \leq y \leq 10 \}$ with surface density $\delta(x, y) = 2x^3y$. The total mass $m$ is calculated by integrating the density over the area $\Omega$: $m = \iint_{\Omega} \delta(x, y) \, dA = \int_0^{20} \int_0^{10} 2x^3 y \, dy \, dx.$

First, let's solve the double integral:

1. Integrate with respect to $y$: $\int_0^{10} 2x^3 y \, dy = 2x^3 \left[ \frac{y^2}{2} \right]_0^{10} = 2x^3 \cdot 50 = 100x^3.$

2. Integrate with respect to $x$: $\int_0^{20} 100x^3 \, dx = 100 \left[ \frac{x^4}{4} \right]_0^{20} = 100 \cdot \frac{20^4}{4} = 100 \cdot \frac{160000}{4} = 4 \times 100 \times 4000 = 1600000.$

So, the mass of the plate is $m = 1,600,000$.

Now, the plate is to be divided into two equal mass parts. The total mass is $1,600,000$, so each part should have a mass of $800,000$.

For part (a), to cut the plate horizontally (constant $y$), we need to solve for $y_c$ where the mass of the region $0 \leq y \leq y_c$ is half the total mass.

For part (b), for a vertical cut (constant $x$), we need to solve for $x_c$ where the mass of the region $0 \leq x \leq x_c$ is half the total mass.

We can handle this further with exact cutting points later.

Problem 4: Lagrange Multipliers

We need to find the extreme values of the function: $f(x, y, z) = 2x^2 + xy + y^2 - 2y + z^2$ subject to the constraint: $g(x, y, z) = y + 2x + 1 = 0.$ To apply the method of Lagrange multipliers, we define the Lagrange function: $\mathcal{L}(x, y, z, \lambda) = 2x^2 + xy + y^2 - 2y + z^2 + \lambda (y + 2x + 1).$

The critical points are found by solving the system of equations given by the gradients:

1. $\frac{\partial \mathcal{L}}{\partial x} = 4x + y + 2\lambda = 0$,
2. $\frac{\partial \mathcal{L}}{\partial y} = x + 2y - 2 + \lambda = 0$,
3. $\frac{\partial \mathcal{L}}{\partial z} = 2z = 0$ (so $z = 0$),
4. $\frac{\partial \mathcal{L}}{\partial \lambda} = y + 2x + 1 = 0$ (the constraint).

We can solve this system to find the critical points and determine which correspond to minima or maxima by testing the values of $f$ at the points.

Problem 5: Double Integral with Changing the Order of Integration

We are given the integral: $I = \int_{-1}^3 \int_0^7 \frac{s^{5/2 - y/2}}{3x} \, dx \, dy.$

To change the order of integration, we first need to analyze the region of integration and rewrite the limits accordingly. After changing the order of integration, we will evaluate the new double integral.

We can proceed with solving this step-by-step and evaluate the integrals explicitly. Would you like to dive deeper into any of these problems for further details or clarifications?

Further Questions:

1. How do you change the order of integration in problem 5 systematically?
2. Can the Lagrange multiplier method in problem 4 be applied to functions with more than one constraint?
3. What methods can be used to simplify the double integral in problem 3?
4. How do you interpret the results of optimization problems in practical terms (e.g., minimum cost)?
5. What are the conditions under which double integrals can be swapped?

Tip:

When solving double integrals, make sure the limits are correctly aligned with the region of integration; changing the order requires careful attention to how the limits change.