Volumes Using CrossSections Solids not generated by Revolution

Volumes Using Cross-Sections Solids not generated by Revolution Solids of Revolution Cross-Sections Sec(6. 2) Volumes Using Cross-Sections Sec(6. 2) The Disk Method The Washer Method Volumes Using Cylindrical Shells Sec(6. 3)

Volumes Using Cross-Sections Volume of a disk

Volumes Using Cross-Sections 1 The Disk Method Strip with small width generate a disk after the rotation

Volumes Using Cross-Sections 1 The Disk Method Several disks with different radius r

Volumes Using Cross-Sections

1 Disk cross-section x step 7 The volume is given by step 1 Graph and Identify the region step 2 Draw a line (L) perpendicular to the rotating line at the point x step 3 Points: boundary of the region and L step 3 Rotate this line. A circle is generated step 4 Find the radius r of the circe in terms of x step 5 step 6 Intersection point between L, curve Now the cross section Area is Specify the values of x Intersection point between L, rotating axis

Intersection point between L, curve Example 2: Disk cross-section x 1) Graph and Identify the region 2) Draw L perpend at location x 3) Points coord: boundary and L 4) Find r for circle in terms of x 5) values of x 0 1 6) volume Check Rotate L. A circle is generated Intersection point between L, rotating axis cross section Area

Volumes Using Cross-Sections

VOLUMES

Solids of Revolution rotate the red line about the x-axis

Volumes Using Cross-Sections Sec(6. 2) The Disk Method The Washer Method

Volumes Using Cross-Sections Sec(6. 1) The Disk Method The Washer Method Examples: Classify

VOLUMES Volume = Area of the base X height

VOLUMES

VOLUMES 2 The washer Method If the cross-section is a washer , we find the inner radius and outer radius

2 The washer Method Example: Washer cross-section x Intersection point between L, boundary 1) Graph and Identify the region 2) Draw L perpend at location x 3) Points coord: bdry&L, rot&L 4) Find rout, rin in terms of x Intersection point between L, boundary 5) values of x 6) volume Check Rotate L. 2 circles are generated cross section Area Intersec pt between L, rotation axis

VOLUMES T-102

VOLUMES

VOLUMES Example: Solution:

VOLUMES 3 Example: The Disk Method (rotation axis about y-axis) If the cross-section is a disk, we find the radius of the disk (in terms of y ) and use

3 The Disk Method (rotation axis about y-axis) Example: Disk cross-section y-axis 1) Graph and Identify the region 2) Draw L perpend at location y 3) Points coord: boundary and L 4) Find r for circle in terms of y 5) values of y 6) volume Check Rotate L. A circle is generated cross section Area Solution:

VOLUMES T-152

VOLUMES 4 Example: The Washer Method (about y-axis or parallel) Solution: Washer cross-section y 1) Graph and Identify the region 3) Draw L perpend at location y 4) Points coord: bdry&L, rot&L 5) Find rout, rin in terms of y 6) values of y 7) volume Check Rotate L. 2 circles are generated cross section Area

VOLUMES T-132

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. Cross-section is DISK Rotated by x-axis or a line parallel to x-axis Rotated by y-axis or a line parallel to y-axis How many points of intersection Cross—section is WASHER

VOLUMES T-111 Remark: before you start solving the problem, read the choices. (d) and (e) are wrong

VOLUMES T-131 Remark: before you start solving the problem, read the choices to figure out which method you use

VOLUMES T-102

VOLUMES BY CYLINDRICAL SHELLS rotating line Parallel to x-axis CYLINDRICAL SHELLS (6. 2) Remarks rotating line Parallel to y-axis rotating line Parallel to x-axis Remarks Using Cross-Section(6. 1) rotating line Parallel to y-axis Cross-section is DISK Cross—section is WASHER SHELL Method

VOLUMES parallel to x-axis SHELLS Cross-Section parallel to y-axis