Find the area of the surface generated by revolving the curve about the x-axis. ,

Question

Find the area of the surface generated by revolving the curve about the x-axis. ​  ,  ​

Quizplus · Accepted Answer

The curve is a semicircle with radius 2, so its area is (1/2)πr² = 2π. To find the area of the surface generated by revolving this curve about the x-axis, we use the formula A = 2π ∫y ds, where y is the equation of the curve and ds is an element of arc length. Since the curve is symmetric about the y-axis, we can only integrate from 0 to 2. To find ds, we use ds = √(1+(dy/dx)²) dx. Since y = √(4-x²), we have dy/dx = -x/√(4-x²), so ds = √(1+(x²/(4-x²))) dx. Thus, A = 2π ∫₀² √(1+(x²/(4-x²))) dx = 2π ∫₀² √((4-x²)/(4-x²)+x²/(4-x²)) dx = 2π ∫₀² √(4/(4-x²)) dx = 2π ∫₀² 2/(2-x) dx = 2π [ln|2-x|]₀² = 2π ln(2). Hence, the area of the surface generated by revolving the curve about the x-axis is 2π ln(2), which corresponds to choice C.

Find the Area of the Surface Generated by Revolving the Curve