Warmup 1) 2) 3)

7.3: Volumes by Slicing Great video on the topic on YouTube http://youtu.be/eRAP87iktNc

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.

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

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.

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

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

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

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

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?

The End