Eroxl's Notes
Disc Integration

Disc integration is a method of finding the volume of a solid of revolution by integrating the volumes of many discs parallel to the axis of rotation.

Function of

If the line that the function is being rotated around is horizontal line (ie. ) then the following equation can be used to calculate the volume of the solid of revolution.

Where is the start of the solid, is the end of the solid, is the horizontal line (ie. if the axis of rotation is then ), and is the function that is being rotated.

Examples

Determine the Volume of Revolution Formed between and when the Curve is Rotated around the line

Y axisX axis001122334455551010Expression 1StartFraction, 4 Over "x" , EndFraction plus "x"Expression 3Expression 4Expression 5Expression 64x+x4x+x

Function of

If the line that the function is being rotated around is vertical line (ie. ) then the following equation can be used to calculate the volume of the solid of revolution.

Where is the start of the solid, is the end of the solid, is the vertical line (ie. if the axis of rotation is then ), and is the function that is being rotated.

Examples

Determine the Volume of Revolution Formed between and when the Curve is Rotated around the \y-axis

Y axisX axis00112211Expression 1Expression 2"y" equals StartFraction, 4 Over pi , EndFraction tangent Superscript, minus 1 , Baseline left parenthesis, "x" , right parenthesisExpression 4y=4πtan−1xy=4πtan−1x