Sec 6 2 VOLUMES Volume Area of the

Sec 6. 2: VOLUMES Volume = Area of the base X height

Sec 6. 2: VOLUMES For a solid S that isn’t a cylinder we first “cut” S into pieces and approximate each piece by a cylinder.

Sec 6. 2: VOLUMES

Sec 6. 2: VOLUMES 1 Disk cross-section x If the cross-section is a disk, we find the radius of the disk (in terms of x ) and use Rotating axis animation

Sec 6. 2: VOLUMES 1 Disk cross-section x use your imagination

Sec 6. 2: VOLUMES 1 Disk cross-section x use your imagination

Sec 6. 2: VOLUMES 1 step 2 step 3 step 4 step 5 Disk cross-section x Intersection point between L, curve Graph and Identify the region Draw a line perpendicular (L) to the rotating line at the point x Intersection point between L, rotating axis Find the radius r of the disk in terms of x Now the cross section Area is step 6 The volume is given by Specify the values of x

Sec 6. 2: VOLUMES

Sec 6. 2: VOLUMES

Sec 6. 2: VOLUMES Volume = Area of the base X height

Sec 6. 2: VOLUMES

Sec 6. 2: VOLUMES 2 washer cross-section x If the cross-section is a washer , we find the inner radius and outer radius

Sec 6. 2: VOLUMES 2 step 1 washer cross-section x Graph and Identify the region step 2 Draw a line perpendicular to the rotating line at the point x step 3 Find the radius r(out) r(in) of the washer in terms of x step 4 step 5 Intersection point between L, boundry Now the cross section Area is Intersection point between L, boundary step 6 The volume is given by Specify the values of x

Sec 6. 2: VOLUMES T-102

Sec 6. 2: VOLUMES Example: Find the volume of the solid obtained by rotating the region enclosed by the curves y=x and y=x^2 about the line y=2. Find the volume of the resulting solid.

Sec 6. 2: VOLUMES 3 Disk cross-section y If the cross-section is a disk, we find the radius of the disk (in terms of y ) and use

Sec 6. 2: VOLUMES 3 step 1 Disk cross-section y Graph and Identify the region step 2 Rewrite all curves as x = in terms of y step 2 Draw a line perpendicular to the rotating line at the point y step 3 Find the radius r of the disk in terms of y step 4 step 5 Now the cross section Area is step 6 The volume is given by Specify the values of y

Sec 6. 2: VOLUMES 4 washer cross-section y If the cross-section is a washer , we find the inner radius and outer radius Example: The region enclosed by the curves y=x and y=x^2 is rotated about the line x=-1. Find the volume of the resulting solid.

Sec 6. 2: VOLUMES 4 step 1 washer cross-section y Graph and Identify the region step 2 Rewrite all curves as x = in terms of y step 2 Draw a line perpendicular to the rotating line at the point y step 3 Find the radius r(out) and r(in) of the washer in terms of y step 4 step 5 Now the cross section Area is step 6 The volume is given by Specify the values of y

Sec 6. 2: VOLUMES T-091

Sec 6. 2: VOLUMES T-072

Sec 6. 2: VOLUMES 4 washer cross-section y If the cross-section is a washer , we find the inner radius and outer radius T-102

Sec 6. 2: VOLUMES SUMMARY: The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. solids of revolution NOTE: solids of revolution Rotated by a line parallel to x-axis ( y=c) Rotated by a line parallel to y-axis ( x=c) The cross section is perpendicular to the rotating line Cross-section is DISK Cross—section is WASHER

Sec 6. 2: VOLUMES not solids of revolution The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. We now consider the volumes of solids that are not solids of revolution.

Sec 6. 2: VOLUMES T-102

Sec 6. 2: VOLUMES T-092

Sec 6. 2: VOLUMES