Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 40

The quantity to be calculated is the area of what is called a surface of revolution. The function y = 3x is rotated about the x-axis and the surface that is created in this way is a surface of revolution. The area to be calculated is definite, since we consider only the region of the x-axis x in [0,3] , that is, x between 0 and 3.
The formula for a surface of revolution (which is an area, A) is given by
A = int_a^b (2pi y) sqrt(1 + (frac(dy)(dx))^2) dx

The circumference of the surface at each point along the x-axis is 2pi y and this is added up (integrated) along the x-axis by cutting the function into tiny lengths of sqrt(1 + (frac(dy)(dx))^2) dx
ie, the arc length of the function in a segment of the x-axis dx in length, which is the hypotenuse of a tiny triangle with width dx and height dy . These lengths are then multiplied by the circumference of the surface at that point 2 pi y to give the surface area of rings around the x-axis that have tiny width dx yet have edges that slope towards or away from the x-axis. The tiny sloped rings are added up to give the full sloped surface area of revolution.
In this case, frac(dy)(dx) = 3 and since the range over which to take the arc length is [0,3] we have a = 0 and b = 3 . Therefore, the area required, A, is given by
A = int_0^3 6 pi x sqrt(10) dx = 3sqrt(10) pi x^2 | _0^3 = 27sqrt(10)pi

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