Matthew V. answered • 04/05/20
Extensive knowledge of physics and mathematics, patient and kind.
The volume of a solid produced by rotating a function of x around the y-axis can be calculated using the shell method. Draw a vertical line from y=3 to y=x^(1/2). Imagine rotating that infinitely thin line of Δx thickness around the y-axis such that it forms an infinitely thin shell of volume ΔV. The volume of this shell is ΔV=2πxyΔx, where x is the distance between the line and the y-axis and y is the height of the line.
To construct an integral that uses this principle of shells to find a full volume, we must first find an interval. Since this shape starts at the y-axis (x=0), one endpoint of the interval is known to be 0. The other can be found by setting 3=x^(1/2), which yields x=9, so the interval is [0,9]. Finally, the y component is, in this case, simply 3-x^(1/2), since we are evaluating the space between these two functions and 3>=x^(1/2) in the increment [0,9].
To integrate ΔV into V, we take the following steps: V = ∫dV = ∫[0,9] 2πxydx = ∫[0,9] 2πx(3-x^(1/2))dx
= ∫[0,9] 6πx-2πx^(3/2)dx, which can be solved as 3πx^2-(4/5)πx^(5/2) [0,9], which comes to 48.6π or 152.7
