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Image: ubuntu2204
Kernel: SageMath 9.2

Volumes of Revolution - A Methodical Approach

var('x y') # usually we're rotating a portion of a function, so the piecewise feature is useful f = piecewise([([0, 1], x^2)]) g = piecewise([([0, 1], sqrt(x))])
p1 = plot(f, 0, 1, gridlines=True) t1 = text("I", (0.8, 0.2), fontsize='large', fontweight='bold') p2 = plot(g, 0, 1) t2 = text("II", (0.2, 0.8), fontsize='large', fontweight='bold') t3 = text("III", (0.4, 0.4), fontsize='large', fontweight='bold') (p1 + t1 + p2 + t2 + t3).show()
Image in a Jupyter notebook

This notebook will demonstrate many permutations of rotating the above regions about various axes using disk, washer and shell methods. Depending on the goal, we will integrate either with respect to x- or with respect to y.

Rotate regions I, II, and III about the x axis.

Region 1 is bounded by the curve y=x2y = x^2 above, the x axis below, and the line x=1x=1. Here is the surface of the volume created when rotated about the x-axis.

revolution_plot3d(f, (x,0,1), parallel_axis='x', show_curve=True, opacity=0.7).show(aspect_ratio=(1,1,1))

Using the disk method, with the disk perpendicular to the x-axis,

V=A  dxV = \int A \; dxA=πr2A = \pi r^2r=f(x)r = f(x)V=π01(f(x))2  dxV = \pi \int_{0}^{1} (f(x))^2 \; dx
pi * integrate(f^2, x, 0, 1)
1/5*pi

Region 2 is bounded by the curve y=xy = \sqrt{x} below, the y axis, and the line y=1y=1 above. Here is the surface of the volume created when rotated about the x-axis.

l = 1 sur1 = revolution_plot3d(l, (x,0,1), parallel_axis='x', opacity=0.5, rgbcolor=(1,0.5,0)) sur2 = revolution_plot3d(g, (x,0,1), parallel_axis='x', show_curve=True, mesh=True, opacity=0.5, rgbcolor=(0,1,0)) (sur1 + sur2).show()