Discussion
Match the application with the integrand it calls for. 1. Arc length 2. Volume of revolution about the x-axis 3. Surface of revolution about the x-axis P. \( \pi y^{2} \) Q. \( \sqrt{1 + y'^{2}} \) R. \( 2\pi y\sqrt{1 + y'^{2}} \)
Answer: A
Arc length integrates the arc element alone. A volume slice is a disc of area \( \pi y^{2} \) and thickness \( dx \). A surface band has circumference \( 2\pi y \) and width equal to the arc element, so it carries both factors, which is what distinguishes it from the volume
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