In this lesson, we will learn how to use integration to find the area of the surface of revolution of a parametrically defined curve.

Q1:

Consider the parametric equations 𝑥=2𝜃cos and 𝑦=2𝜃sin, where 0≤𝜃≤𝜋.The area of the surface 𝑆 obtained by rotating this parametric curve 2𝜋 radians about the 𝑥-axis can be calculated by evaluating the integral 2𝜋𝑦𝑠d where dddddd𝑠=𝑥𝜃+𝑦𝜃𝜃.

Find d𝑠.

Hence, find the surface area of 𝑆 by evaluating the integral.

Q2:

Consider the parametric equations 𝑥=2𝑡−1 and 𝑦=𝑡+1, where 0≤𝑡≤2. Calculate the area of the surface obtained when the curve is rotated 2𝜋 radians about the 𝑥-axis.

Q3:

Determine the surface area of the solid obtained by rotating the parametric curve 𝑥=1+2𝑡 and 𝑦=1−2𝑡, where 0≤𝑡≤2, about the 𝑦-axis.

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