A smooth curve C is defined analytically by the equation for all in its domain
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
A smooth curve C is defined analytically by the equation x=4−y for all y in its domain. A region R in the first quadrant is bounded on the right by C, on the left by the y–axis, and above by the horizontal line y=4.
(1) Verify algebraically that C intersects the y–axis exactly once and state the y–coordinate of that intersection.
(2) Using only information provided in the prompt, determine the greatest and least y–values that belong to region R, and explain how you know.
(3) Write (but do not evaluate) a single definite integral whose value equals the area of region R. Clearly indicate the limits of integration and justify the sign of the integrand throughout the interval.
(4) Describe, without performing any additional algebra, whether the integrand you wrote in part (c) is non-negative, non-positive, or changes sign on the interval of integration, and explain the significance of this fact in the context of finding an area.