6.2 - Volumes

Definition: Right Cylinder Let B 1 and B 2 be two congruent bases. A cylinder is the points on the line segments perpendicular to the bases that join B 1 and B 2. B1B1 B2B2 Alternate Definition A cylinder is a surface whose cross section in every plane parallel to a given plane are the same.

Volumes of Solids P x is perpendicular to the x axis. A(x) is a function that represents the cross-sectional area where a ≤ x ≤ b.

Volumes of Solids Establish n subintervals on [a, b] and let Δx. Choose the midpoint to be x i. Then a cylinder is formed with volume V(S i ) = A(x i ) Δx

Estimate of the volume of the figure using n = 7 subintervals. The volume can be approximated by

Definition of Volume Let S be a solid that lies between x = a and x = b. If the cross- sectional area of S in the plane P x, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is V = Area × Width

Volumes of Solids of Revolution: Example Let R be the region bounded by y = 4 – x 2 and y = 0. Find the volume of the solids obtained by revolving R about the y-axis. 2-D Graph What is the function that represents the radius? 3-D Graph What is the function that represents the area of the disk?

It will be easier to integrate along the y-axis since it is perpendicular to the disks. Thus, the radius, x, is the distance from the y-axis to the function or The area of the disk is given by The volume of revolution is given by

Volumes of Solids of Revolution: Example Let R be the region bounded by the graphs of y = x 2, x = 0, and y = 1. Compute the volume of the solid formed by revolving R about the x- axis. 2-D Graph What is the function that represents the radius? 3-D Graph What is the function that represents the area of the washer?

Now we need to determine the area function for the washer.

The volume can now be found by

Volumes of Solids of Revolution: Example Let R be the region bounded by the graphs of y = x 2, x = 0, and y = 1. Compute the volume of the solid formed by revolving R about the line y = 2. 2-D Graph What is the function that represents the radius? 3-D Graph What is the function that represents the area of the washer?

Now we need to determine the area function for the washer. The volume is given by