Exercise 3.7 Class 12 Math Solutions - Unit 3 Integration, Punjab Board
Step-by-step solutions to Exercise 3.7, Unit 3: Integration, for Class 12 Punjab Board students. Covers real-world applications of integration including moments of inertia (uniform rod and rectangular lamina), rectilinear kinematics, average function values, and pharmacokinetic drug dosage AUC calculations, worked out question by question.
Exercise Overview
Exercise 3.7 serves as the comprehensive capstone for Unit 3: Integration in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum, connecting integral calculus directly to real-world physics, mechanics, and biosciences. In rotational mechanics, students derive the Moment of Inertia (I = ∫ r² dm) for continuous bodies, specifically evaluating a uniform thin rod of length (2a) about an axis through its center of mass and perpendicular to its length (I = (1/3)Ma²), alongside uniform rectangular laminas.
In kinematics, the exercise explores linear particle motion where velocity is the antiderivative of acceleration (v(t) = ∫ a(t) dt + v₀) and displacement is the antiderivative of velocity (s(t) = ∫ v(t) dt + s₀). Furthermore, students compute the Average Value of a Continuous Function over ([a, b]) using (f_avg = (1/(b - a)) ∫ₐᵇ f(x) dx), and apply pharmacokinetic Area Under the Curve (AUC) integration to calculate total drug exposure and required therapeutic dosage from plasma concentration curves (C(t)).
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 3.7 provides high-yield applied questions that test conceptual understanding and multi-step derivations. These step-by-step solutions provide rigorous calculus setups, explicit boundary conditions, and clean evaluations for every problem.