Region S in the first quadrant is defined as follows: • For , S is the region under…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region S in the first quadrant is defined as follows:
• For 0≤x≤1, S is the region under the curve y=x and above the x–axis.
• For 1≤x≤4, S is the region between the curves y=3 and y=x.
Region S is revolved about the x–axis.
(a) Determine on which subintervals of 0≤x≤4 the resulting cross-sections are disks and on which subintervals they are washers. Provide reasoning that references the inner radius on each subinterval.
(b) Construct a complete, piecewise-defined definite integral expression that represents the volume of the solid obtained by revolving S about the x–axis. Clearly label each part of your expression and include appropriate limits of integration.
(c) Evaluate your piecewise integral to obtain the exact volume of the solid.