Area Under and Between Curves Volumes of Solids of Revolution Myndert Papenhuyzen

Section I: Area Under and Between Curves

General Equations Where f(x) or f(y) are the upper / right function, and g(x) or g(y) are the lower / left function. REMEMBER: if an axis is involved, then substitute 0 (zero) for the appropriate function.

Example 1 Find the area enclosed by... The graph f(x) = x², the x-axis, and the lines x = 1 and x = 2.

f(x) = x² between x=1 and x=2 2 2

Example 2 Find the area enclosed by... The graph f(x) = x³ − x, the x-axis, and the lines x = −1 and x = 1.

f(x) = x³ − x between x = −1 and x = 1 2

Example 3 Find the area enclosed by... The graphs f(x) = x³ and g(x) = 4x.

Enclosed by f(x) = x³ and g(x) = 4x 2

On Your Own Find the area of the region enclosed by... f(y) = y³ – y and g(y) = 1 – y 4

Section II: Volume of Solids of Revolution

Technique 1: Disk Method

The disk method should be used when the region to be revolved touches the line of revolution completely (i.e. the line acts as one of the bounding functions), and the line of revolution is in the same variable of the bounding function (e.g. a y = function is revolved around a y = line).

Disk Example Find the volume of the area bounded by... The graph and the lines x = 0 and x = 4 … revolved around the x-axis.

from [0, 4] around x-axis

Technique 2: Washer Method Where function f is the outer function when revolving and function g is the inner function.

Technique 2: Washer Method The washer method should be used when the region to be revolved does not completely touch the line of revolution, and the line of revolution is in the same variable as the region's bounding functions.

Washer Example Find the volume of the region enclosed by... The graphs f(x) = sin(x) and g(x) = cos(x)...rotated around the x-axis.

Washer Example

Technique 3: Shell Method

The shell method should be used when the line of revolution is NOT in the same variable as the bounding functions (e.g. the region is bounded by y= functions, but revolved around an x= line).

Shell Example Find the volume of the region enclosed by... The graph f(x) = ln x, the x-axis, and the line x = e...revolved around the y-axis.

f(x) = ln x between [1, e] around y- axis

On Your Own Find the volume of the region bounded by... The graph f(x) = x and the lines y = 0, x = 2, and x = 4... and revolved around the x-axis.

f(x) = x, y = 0 from [2, 4] around x-axis

Modification 1: Non-Axial Revolution When rotating a region around a line that is not an axis, then the region's bounding functions must be shifted so that the line coincides with the appropriate axis.

Non-Axial Revolution Example 1 Find the volume of the region enclosed by... The graph f(x) = x² and lines y = 0, x = −1, and x = 1....revolved around the line y = −1.

f(x) = x² between [−1, 1] around y = −1

Non-Axial Revolution Example 2 Find the volume of the region enclosed by... The graphs f(x) = x³ and g(x) = x 1/2...revolved around the line y = 2.

f(x) = x³ and g(x) = x 1/2 around y = 2

On Your Own Find the volume of the region bounded by... The graph f(x) = 1/x² and the lines y = 0, x = 1, and x = 3... and revolved around the line y = -1

f(x) = 1/x², y = 0 from [1, 3] around y = -1

f(x) = 1/x², y = 0 from [1, 3] around y = −1

Modification 2: Cross-Sections When a region is described as a side of a solid whose cross-sections are a certain shape, then the volume of the solid is equal to the integral of the area of the cross-section.

Cross Section Example The region bounded by... The graph f(x) = x² and the lines y = 0 and x = 4...is a face of a solid, whose cross-sections perpendicular to the x-axis are squares. Find the volume of the solid.

f(x) = x², y = 0, x = 4

Bibliography / Sources Music: Spring, mvt. I (Vivaldi) and Summer, mvt. III (Vivaldi), as recorded by John Harrison. The music is licensed under CC-BY-SA, which allows for free use and adaptation, as long as this license is maintained with the work.CC-BY-SA Images: Graph images generated by MathGV and Lybniz Graph Plotter. Equation images generated by the CodeCogs Equation Editor.CodeCogs Equation Editor Information: The information is taken from in-class notes and examples, along with some exercises from James Stewart's Calculus, fifth edition

© 2011, Myndert Papenhuyzen