1. Warmup 1) 2) 3)

2. 7.3: Volumes by Slicing Great video on the topic on YouTube

4. If the area a cross section of a solid is know
and can be expressed in terms of x, then the volume
of a typical slice can be determined.
The volume can typically be obtained by letting
the number of slices increase indefinitely.

5. 3
Find the volume of the pyramid:
Consider a horizontal slice through the pyramid.
The volume of the slice is s2dh.
If we put zero at the top of the pyramid and make down the positive direction, then s=h. h s
This correlates with the formula: dh 3

6. Method of Slicing: 1
Sketch the solid and a typical cross section.
Find a formula for V(x).
(Note that I used V(x) instead of A(x).) 2 3
Find the limits of integration. 4
Integrate V(x) to find volume.

7. h 45o x A 45o wedge is cut from a cylinder of radius 3 as shown.
Find the volume of the wedge.
You could slice this wedge shape several ways, but the simplest cross section is a rectangle. x y
If we let h equal the height of the slice then the volume of the slice is: x h
45o
Since the wedge is cut at a 45o angle:
Since

8. Even though we started with a cylinder, p does not enter the calculation!x y

9. p Cavalieri’s Theorem:
Two solids with equal altitudes and identical parallel cross sections have the same volume.
Identical Cross Sections p

10. Group:
A solid has as its base a circle
and all its cross sections parallel to the y-axis are squares.
Find the Volume of the solid.
144

11. Group:
The base of a solid is a region bounded by the
parabola and the line y = 4
and each plane section perpendicular to the y-axis
is an equilateral triangle. The volume of the solid is?

12. The End