Solids of Revolution: Volumes by Disks
A solid of revolution is a solid three-dimensional figure created by rotating a subregion of the xy-plane around a straight line, usually the x-axis or the y-axis. Assuming there is a single function y=fx which defines the upper boundary of this region, and the lower boundary is the x-axis, and the region is bounded on the left and right by the lines x=a and x=b, the volume of the solid can by found using the disk method.
Rotation around the x-axis
Given a curve y=fx that is continuous on x = a, b, we can divide a, b into n subintervals xi , xi+1, each of width Δ x.
Then, we can form a rectangle from the x-axis to the curve, of width Δ x and height fx.
Rotating this rectangle about the x-axis, a vertical disk is formed with radius fxi∗ and width Δ x, where xi∗∈xi , xi+1.
The volume of this disk is given by Vdisk = π⁢fxi∗2⁢Δ⁢x. The volume of the entire solid can be found by summing the volumes of all n disks, while letting them become infinitely thin by having n →∞.
Therefore, the volume of the solid of revolution is:
Vsolid = limn→∞∑i=1nπ⁢fxi∗2⁢Δ⁢x = ∫abπ⁢fx2 ⅆx
Choose a function from the drop down menu, enter a function in the textbox and press "Enter", or draw a curve in the plot on the left. The solid of revolution will be drawn in the plot on the right, and its volume will be shown below the plot. Click "Animate" to see the solid being created by revolving it about the x-axis.
Custom graphy = sin(x)y = xy = x^2y = x^3y = sqrt(x)y = exp(x)y = 5 fx=
Volume of Solid:
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