The region in the –plane is bounded above by the curve
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
The region R in the xy–plane is bounded above by the curve y=ln(x+1), bounded below by the x–axis, and bounded on the left by the y–axis. The region is revolved about the horizontal line y=−2 to create a three–dimensional solid.
(a) For a fixed value of x with 0≤x≤3, express the distance from the line y=−2 to the curve y=ln(x+1) and the distance from the line y=−2 to the x–axis. Explain why each expression is appropriate for describing radii of the circular cross–sections produced at position x. Your explanation should make clear which distance serves as the outer radius and which serves as the inner radius.
(b) Identify the lower and upper limits of x that must be used in a definite integral to compute the volume of the solid. Justify your choices by referencing geometric features of the region R. No evaluation of the integral is required.