Specialist2025 Exam 2 Section A · Q9CAS active1 mark

Surface area of a parametric curve rotated about the x-axis

A parametric curve is given by $x = kt$ , $y = e^{kt}$ , where $k$ is a positive constant. The curve is rotated about the -axis from to , where , to form a surface of revolution.

The area of this surface is given by

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2\pi\int_{a}^{b} e^{kt}\sqrt{k^{2}t^{2}+e^{2kt}}\,dt

2\pi\sqrt{k}\int_{a}^{b} e^{kt}\sqrt{1+e^{kt}}\,dt

2\pi\int_{e^{ka}}^{e^{kb}} \sqrt{1+u^{2}}\,du

2\pi\int_{e^{ka}}^{e^{kb}} u\sqrt{1+u^{2}}\,du

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