Exercise 3.6 Class 12 Math Solutions - Unit 3 Integration, Punjab Board
Step-by-step solutions to Exercise 3.6, Unit 3: Integration, for Class 12 Punjab Board students. Covers real-world applications of definite integrals including area under and between curves, Consumer and Producer Surplus in economic market equilibrium, and volumes of solids of revolution by cross-sectional slicing, worked out question by question.
Exercise Overview
Exercise 3.6 concludes Unit 3: Integration in Class 12 Mathematics by presenting the major applied disciplines of definite integral calculus under the Punjab Board PCTB / PECTAA curriculum. The first geometric application is the calculation of Area Under the Curve (AUC) and Area Bounded Between Curves. Students learn how the definite integral (∫ₐᵇ f(x) dx) evaluates the net signed area, requiring piecewise absolute value integration (∫ₐᵇ |f(x)| dx) when curves cross below the x-axis.
The second segment introduces mathematical economics, focusing on Consumer Surplus (CS) and Producer Surplus (PS) at the market equilibrium price and quantity (x₀, p₀) where demand equals supply (D(x₀) = S(x₀)). Students apply formulas with respect to quantity (CS = ∫₀ˣ⁰ D(x) dx - p₀x₀) and (PS = p₀x₀ - ∫₀ˣ⁰ S(x) dx) and price to quantify market welfare benefits. Finally, the exercise explores Solids of Revolution, employing the cross-sectional disk method to determine the volume generated by rotating a planar region around the coordinate axes using (V = π ∫ₐᵇ [f(x)]² dx).
For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 3.6 is a prime source of high-scoring Section C Long Questions and practical application problems. These step-by-step solutions provide explicit integration steps, clean diagrams, and complete economic and geometric evaluations.