# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - Definite integrals for calculating areas under curves and between curves - Integration with respect to both $x$ and $y$ (horizontal and vertical slices) - Volumes of solids of revolution (Disk and Washer methods) - Applications to physics: Work, force, and Hooke's Law for springs - Applications to economics: Consumer surplus and producer surplus - Total accumulation from rate of change (revenue, motion) - Position functions and rectilinear motion --- # **Important Formulas** Below are the key formulas used in this exercise: **Area Formulas:** $$A = \int_a^b f(x) \, dx \quad \text{(Area under curve)}$$ $$A = \int_a^b [f(x) - g(x)] \, dx \quad \text{(Area between curves)}$$ $$A = \int_c^d [f(y) - g(y)] \, dy \quad \text{(Area using horizontal slices)}$$ **Volume Formulas (Solids of Revolution):** $$V = \pi \int_a^b [f(x)]^2 \, dx \quad \text{(Disk Method)}$$ $$V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right) \, dx \quad \text{(Washer Method)}$$ **Physics Applications:** $$\text{Hooke's Law: } F = kx$$ $$W = \int_a^b F(x) \, dx \quad \text{(Work by variable force)}$$ $$W = \frac{1}{2}kx^2 \quad \text{(Work to stretch spring)}$$ **Economic Applications:** $$\text{Consumer Surplus: } CS = \int_0^{q^*} D(q) \, dq - p^*q^*$$ $$\text{Producer Surplus: } PS = p^*q^* - \int_0^{q^*} S(q) \, dq$$ **Accumulation from Rate:** $$\text{Total} = \int_a^b r(t) \, dt$$ --- # **Summary** This exercise comprehensively covers applications of definite integration across geometry, physics, and economics. Geometrically, it emphasizes calculating areas between curves (using both vertical and horizontal slices) and finding volumes of solids of revolution using disk and washer methods. Physically, it applies integration to calculate work done by variable forces, particularly in stretching springs using Hooke's Law. Economically, it covers consumer and producer surplus calculations and determining total revenue from marginal rate functions. Key strategies include carefully determining integration limits from intersection points, selecting the appropriate variable of integration to simplify calculations, and properly identifying radii for volume calculations.