A storage tank is formed by revolving a planar region about the -axis

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A storage tank is formed by revolving a planar region about the yy-axis. The region is bounded on the right by the curve x=3+2cos⁡(y2)x=3+2\cos\left(\dfrac{y}{2}\right), on the left by the yy-axis (x=0x=0), below by y=0y=0, and above by y=πy=\pi. All dimensions are in meters. (a) Explain why the distance from the yy-axis to a cross-section of the tank at height yy is 3+2cos⁡(y2)3+2\cos\left(\dfrac{y}{2}\right) meters. (b) Write a definite integral that represents the volume of the tank. Do not evaluate yet. (c) Evaluate the integral obtained in part (b) to find the exact volume of the tank in cubic meters. (d) Interpret the physical meaning of the factor π\pi that appears in your final answer and discuss how the periodic nature of the cosine term influences the tank’s shape.

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