Certainly, I can help with each of these problems! Here is an outline of the solutions and an explanation for each part. I'll go through each problem in detail below:

Problem 1: Double Integrals over Rectangles

The goal is to find the volume $V$ under the surface $z = f(x, y) = x \ln(x^2) \sin(y)$ over the rectangle $[2, 4] \times [0, \pi]$.

(a) Estimate $V$ using a Riemann sum

• Step: Divide the rectangle $[2, 4] \times [0, \pi]$ into 6 equal-sized sub-rectangles.
• Divide $x$-interval into 3 equal parts, and $y$-interval into 2 equal parts.
• Evaluate $f(x, y)$ at the bottom-left corner of each sub-rectangle, then sum them to approximate the volume.

(b) Set up an iterated double integral for $V$

• Integral Setup: $V = \int_{0}^{\pi} \int_{2}^{4} x \ln(x^2) \sin(y) \, dx \, dy$

(c) Evaluate the integral

• Solution: Solve the integral step-by-step, integrating with respect to $x$ first, then $y$.

Problem 2: Set up an iterated integral

The domain $D$ is bounded by $y = x^2$ and $x + 2y = 6$.

• Step: Determine the limits for $x$ and $y$ based on the boundaries.
• Integral Setup: $\int \int_{D} (6x + 3y) \, dA$ with appropriate bounds on $x$ and $y$ based on the intersections of $y = x^2$ and $x + 2y = 6$.

Problem 3: Volume of the Solid above a Triangular Domain

Given a triangular domain in the $(x, y)$-plane with vertices $(0, 0), (2, 0), (0, 1)$, find the volume of the solid above $D$ and below the plane $2x + 4y + z = 8$.

• Rewrite the plane equation as $z = 8 - 2x - 4y$.
• Set up a double integral for volume: $V = \int \int_{D} (8 - 2x - 4y) \, dA$
• Determine the limits based on the triangular region $D$.

Problem 4: Calculate the Integral by Reversing Order of Integration

The integral given is: $\int_{x=0}^{x=2} \int_{y=x}^{y=2} e^{-4y^2} \, dy \, dx$

• Reverse the order of integration to evaluate it more easily.
• Rewrite the bounds accordingly, then integrate.

Would you like me to go through each of these solutions in more detailed steps, including full calculations? Let me know if you need specific help on any part of the problems above.

Questions to Deepen Understanding

1. How do we determine the appropriate sub-rectangle size for Riemann sums when approximating double integrals?
2. What is the significance of changing the order of integration, and when is it most useful?
3. How do we interpret the boundaries for regions defined by curves, like $y = x^2$ and $x + 2y = 6$?
4. Why do we need to rewrite a plane equation in terms of $z$ when calculating volume?
5. How can we verify the results obtained from double integrals?

Tip:

When dealing with complex regions for integration, sketching the region and identifying the bounds visually can make setting up integrals much easier.