Week 9: Signed Area and Optimization
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# Week 9: Signed Area and Optimization > **Course:** Jan 2026 - Mathematics I > **Focus:** Interpreting sign in area, then turning geometry into an optimization problem. --- ## 1.
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Week 9: Signed Area and Optimization
Course: Jan 2026 - Mathematics I Focus: Interpreting sign in area, then turning geometry into an optimization problem.
1. Core idea
Week 9 mixes definite integrals with geometry. The same curve can contribute positive or negative area depending on where it sits relative to the x-axis.
What to remember
- Signed area counts above-axis and below-axis regions differently.
- Volume problems often reduce to one variable using a geometric constraint.
- For an inscribed cylinder, the sphere relation gives the main substitution.
- Maxima usually come from differentiating the reduced expression and checking the critical point.
2. Common traps
- Confusing signed area with absolute area.
- Forgetting that optimization problems need a constraint before differentiation.
- Dropping the square-root relation when connecting radius and height.
3. Speedrun pattern
When you see a Week 9 question, ask:
- Is the quantity signed or unsigned?
- What is the one-variable constraint?
- What is the cleanest point to differentiate?