MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
In the first quadrant, the region enclosed by the line y=x and the quarter-circle x2+y2=4 is revolved about the x–axis, producing a solid Q.
(a) Find the x-coordinate of the point where the line y=x intersects the quarter-circle x2+y2=4 in the first quadrant.
(b) Formulate a definite integral that represents the volume of Q. Clearly state the expressions for the outer radius R(x) and inner radius r(x) used in your integral.
(c) Evaluate the integral from part (b) to obtain the exact volume of Q. Show all algebraic steps, including any necessary substitutions.
(d) Suppose the region described above represents the cross-section of a storage vessel (measured in meters). Interpret the numerical value of the volume you found in part (c) in terms of the vessel’s capacity, including correct units.