Disc Method

Disc Method is a method for calculating the Volume of a 3D shape by rotating a 2D shape.

The strategy of this method is:

• First to ROTATE an infinitely small piece of the whole graph
• Calculate the AREA of this Rotated Circle, or so called disc.
• Integrate all the discs.

►Jump to Khan academy for some practice: Disc Method
▼Refer to the article: Finding volumes of 3-D objects with circular symmetry in at least one dimension

Example

Solve:

• First need to completely understand the question and visualize it using Disc Method:

• Calculate the area of disc and integrate them:

Example

Solve:

• The radius of the disc is exactly equal to y value, means r = y = eˣ
• The area of the disc then be Area(x) = πr² = π · e²ˣ
• Since We’re integrating Horizontally along X-axis, so the integral should be ʃ A(x) dx
• Integral all the discs: Volume = ʃ A(x) dx = ʃ π · e²ˣ dx
• Calculate the integral.

Example

Solve:

• The 2D shape is like this one:

• So we are to integrate discs along X-axis: ʃ Area(x) dx
• The interval is [0, 4].
• It’s tricky to get the radius of the disc: r = y - 1 = √x +1 -1 = √x
• So the area of each disc is: Area(x) = πr² = πx
• Integrate those discs over inteval [0, 4]: Volume = ʃ Area(x) dx = ʃ πx dx = 8π.