The document contains a comprehensive final practice set for Math225 focusing on integral calculus. Here’s a summary of its contents:

1. u-Substitution
   A set of definite integrals requiring the application of u-substitution.

2. Applications of Anti-derivative

   • Prove a formula for displacement under constant acceleration.
   • Solve a motion problem involving a ball thrown upward, determining height, motion direction, time to hit the ground, and maximum height.

3. Area Problems
   Problems involve sketching regions enclosed by curves and calculating their area.

4. Volume Using Disk/Washer Method
   Problems ask for setting up integrals (but not evaluating them) to find volumes of solids generated by rotating regions about specified lines.

5. Volume Using Cylindrical Shell Method
   Similar to the previous, but using the cylindrical shell method for setting up the volume integrals.

6. Average Value of a Function
   Problems involve finding the average value of functions over given intervals.

7. Finding Specific c Values
   For a given average function value, identify $c$ in the interval such that $f_{avg} = f(c)$.

8. Optimization Problems

   • A fencing problem to minimize cost.
   • A box design problem to minimize production costs.

Would you like assistance with solving any specific problem or detailed explanation on a topic from this set?

Expanding Topics:

1. How to solve u-substitution integrals effectively.
2. Analyzing motion under constant acceleration.
3. Techniques for sketching regions bounded by curves.
4. Comparing disk/washer and cylindrical shell methods for volume.
5. Practical applications of optimization problems in real-world scenarios.

Tip: For optimization problems, always start by identifying variables and writing a single equation relating cost or area to those variables.