Section 7 2 Volume The Disk Method Examples

Formulas: Disk Method (about x axis) Note: Function must be of the form: y

Formulas: Disk Method (about y axis) Note: Function must be of the form: x

Formulas: Washer Method (about x axis) Note: Function must be of the form: y

Formulas: Washer Method (about y axis) Note: Function must be of the form: x

Washer Method Revolving about a Line • R(x) = Line - Farthest Graph •

1. Set up and evaluate the integral that gives the volume of the solid

2. Set up and evaluate the integral that gives the volume of the solid

3. Set up and evaluate the integral that gives the volume of the solid

4. Set up and evaluate the integral that gives the volume of the solid

5. Find the volumes of the solids generated by revolving the regions bounded by

6. Find the volume of the solid generated by revolving the region bounded by

7. Find the volume of the solid generated by revolving the region bounded by

8. Find the volume of the solid generated by revolving the region bounded by

9. Find the volume of the solid generated by revolving the region bounded by

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Section 7. 2 Volume: The Disk Method

Examples (from mathdemos. org)

Formulas: Disk Method (about x axis) Note: Function must be of the form: y = _____ and a and b are in terms of x values

Formulas: Disk Method (about y axis) Note: Function must be of the form: x = _____ and c and d are in terms of y values

Formulas: Washer Method (about x axis) Note: Function must be of the form: y = _____ and a and b are in terms of x values

Formulas: Washer Method (about y axis) Note: Function must be of the form: x = _____ and c and d are in terms of y values

Washer Method Revolving about a Line • R(x) = Line - Farthest Graph • r(x) = Line - Nearest Graph

1. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis (similar to p. 465 #1 -5)

2. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis (similar to p. 465 #1 -5)

3. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis (similar to p. 465 #7 -9)

4. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis (similar to p. 465 #7 -9)

5. Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines (similar to p. 465 #11 -13) a) b) c) d) the x-axis the y-axis the line x = 8 the line x = 10

6. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 6 (similar to p. 465 #15 -17)

7. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 5 (similar to p. 465 #19 -21)

8. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis (similar to p. 465 #23 -29)

9. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis (similar to p. 465 #33 -35)