1. Solids not generated by Revolution
Volumes Using Cross-Sections
Solids
Solids of Revolution
Solids not generated by Revolution

2. Volumes Using Cross-Sections
The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid
Example: 2
Cross-sections:
Base:
is bounded by the curve
and the line y =2
If the cross-sections of the solid perpendicular to the y-axis are squares

3. VOLUMES
the cross-sections of the solid perpendicular to the y-axis are squares
The base of a solid is bounded by the curve and the line y =2

4. VOLUMES Example: Base: Cross-sections:
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
Base:
is bounded by the curve
and the line x =9
Cross-sections:
If the cross-sections of the solid perpendicular to the x-axis are semicircle

5. VOLUMES Example: Base: Cross-sections:
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
Base:
is bounded by the curve
and the line x =9
Cross-sections:
If the cross-sections of the solid perpendicular to the x-axis are semicircle

6. VOLUMES Example: Base: Cross-sections:
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
Base:
is bounded by the curve
and the line x =9
Cross-sections:
If the cross-sections of the solid perpendicular to the x-axis are semicircle

7. VOLUMES Example: Base: Cross-sections:
The base of a solid is bounded by the curve and the line y = 0 from x=0 to x=pi.
If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles.
Example:
Base:
is bounded by the curve
and the line y =0
Cross-sections:
If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles

8. Volumes Using Cross-Sections
step1
Graph and Identify the region ( graph with an angle)
The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid
Example: 2
step2
Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem)
step3
Find the length (S)of the segment from the two intersection points with the boundary
Cross-sections:
If the cross-sections of the solid perpendicular to the y-axis are squares
step4
Cross-section type:
Square  S = side length
Semicircle  S = diameter
Equilatral  S = side length
step4
Cross-section type:
Square 
Semicircle 
Equilatral 
step6
Specify the values of x
The volume is given by
step7

9. VOLUMES Example: Cross-sections:
The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid
Example:
step1
Graph and Identify the region ( graph with an angle)
step2
Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem)
Cross-sections:
step3
Find the length (S)of the segment from the two intersection points with the boundary
If the cross-sections of the solid perpendicular to the x-axis are semicircle
step4
Cross-section type:
Square  S = side length
Semicircle  S = diameter
Equilatral  S = side length
step4
Cross-section type:
Square 
Semicircle 
Equilatral 
step6
Specify the values of x
The volume is given by
step7

10. VOLUMES
T-102
Type equation here.Type equation here. here. 

11. VOLUMES
T-132

13. VOLUMES
T-122

14. VOLUMES
T-092

15. VOLUMES
54.The base of a solid is a circular disk with radius r = 2. Parallel cross-sections perpendicular to the base are squares.

16. VOLUMES