MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
In the plane, consider the closed region Q that satisfies the following conditions:
• 0≤y≤1;
• x=0 is the left boundary;
• The right boundary is the circle (x−1)2+y2=1.
(a) Solve (x−1)2+y2=1 for x in terms of y, clearly indicating all branches and their respective domains.
(b) For each y in the interval 0≤y≤1, determine which branch of the circle serves as the right boundary of Q. Justify your answer analytically.
(c) Set up all definite integral(s) with respect to y whose total equals the exact area of Q, showing the limits of integration for each integral and the integrand you will use.
(d) Evaluate the integral expression from part (c) to obtain the exact area of Q.