Rotate Curve: Find Surface Area of Resulting Solid

A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around a straight line in its plane (the axis).[1]

Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated about any diameter generates a sphere of which it is then a great circle, and if the circle is rotated about an axis that does not intersect the center of a circle, then it generates a torus which does not intersect itself (a ring torus).
When the axis of rotation is the x-axis and provided that y(t) is never negative, the area is given by[4]

A_x = 2 \pi \int_a^b y(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt.
If the curve is described by the function y = f(x), a ≤ x ≤ b, then the integral becomes

A_x = 2\pi\int_a^b y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx
for revolution around the x-axis, and

A_y =2\pi\int_a^b x \sqrt{1+\left(\frac{dx}{dy}\right)^2} \, dy
for revolution around the y-axis (Using a ≤ y ≤ b). These come from the above formula.

[MrYusufgovani]

Thanks Chris. Your explanation was clear, and I could understand it thoroughly.

[Matchbox]

Can you make some instructional videos on integrateing radical equations, i like add them to my notes. Thanks Dr. Tisdell