Volume of Solid of Revolution


Exercises for students
Sketch the following on paper and check your answers using the above applet:

Sketch the region enclosed by the curve y = (x - 2)^(3/2) for 2 < x < 4, the line x = 4 and the x-axis.
Rotate the region through 4 right angles about the x-axis.

Sketch the region enclosed by the curve x = sqrt(y) for 0 < x < 1 and the curve x = sqrt(2 - y) for 1 < x < 2.
Rotate the region through 4 right angles about the y-axis.

Sketch the region enclosed by the curve y = 9 - x^2 for -3 < x < 3, and the x-axis.
Rotate the region through 4 right angles about the x-axis.

Sketch the region enclosed by the curve x = sqrt(9-y) for 0 < y < 9, and the x- & y-axes.
Rotate the region through 4 right angles about the y-axis.

Sketch the region enclosed by the curve x = 2 + cos(y) for -4.5 < x < 0, the line y=-4.5 and the x- & y-axes.
Rotate the region through 4 right angles about the y-axis.

Sketch the region enclosed by the curve x = sqrt(y), the line y = 2, and the y-axis.
Rotate the region through 4 right angles about the y-axis.

Sketch the region enclosed by the curve x = sqrt(4 - 2y) for 0 < x < 2 and the curve x = sqrt(4 - 4y) for 0 < x < 1.
Rotate the region through 4 right angles about the y-axis.

Summary

Right-click here to download a summary of Areas & Volumes of Revolution.

When rotated about the x-axis, volume of each disc = pi y² δx
Therefore volume of all the discs = Σ pi y² δx
Therefore volume of solid of revolution about x-axis = pi ∫ y² dx

When rotated about the y-axis, volume of each disc = pi x² δy
Therefore volume of all the discs = Σ pi x² δy
Therefore volume of solid of revolution about y-axis = pi ∫ x² dy



You can define the function using the following operators:

+	-	*	/	^
abs( )	sqrt( )	ln( )	exp( )	pi
	sin( )	cos( )	tan( )
	asin( )	acos( )	atan( )
	sinh( )	cosh( )	tanh( )

Right-click here to download this page, the Java Class File and unzip the Javathings Math Package.
This applet uses the com.javathings.math package developed by:
Patrik Lundin
patrik@javathings.com
http://www.javathings.com