1. 8-3 Volumes

2. *We find volumes in the same manner that we found areas – by slicing the totality up and adding the parts together.
*Each “slice” will have a volume – it can be best described as a very short “cylinder.”
Riemann Sum
Def: The volume of a solid of a known integrable cross-section area A(x) from x = a to x = b is the integral of A from a to b.

3. How to Find Volume by Slicing
(1) Sketch solid and a “typical” cross section
(2) Find formula for A(x)
(3) Find limits of integration
(4) Integrate A(x) to find the volume

4. Ex 1) A pyramid 3 m high has congruent triangular sides and a square base that is 3 m on each side. Each cross section of the pyramid parallel to the base is a square. Find the volume of the pyramid.
Area of square = x2
Limits: 0 & 3

5. Area Formulas from Geometry

6. (Let’s just solve 1a. We can simply set up the other three)
HW #1 The solid lies between planes perpendicular to the x-axis at x = –1 and x = 1. The cross sections perpendicular to the x-axis between the planes run from the semicircle
a) The cross sections are circular disks with diameters in the xy-plane.
Area of Circle = r2
Limits: –1 to 1 (Or 0 to 1 if 2)

7. (Just the set up! Not final answer!)
b) The cross sections are squares with bases in the xy-plane.
Area = s2
s = diameter
c) The cross sections are squares with diagonals in the xy-plane.

8. (Just the set up! Not final answer!)
d) The cross sections equilateral triangles with bases in the xy-plane.

9. Ex 7) A mathematician has a paperweight made so that its base is the shape of the region between the x-axis and one arch of the curve y = 2sin x (linear units in inches). Each cross section cut perpendicular to the x-axis (and hence to the xy-plane) is a semicircle whose diameter runs from the x-axis to the curve. (Think of the cross section as a semicircular fin sticking up out of the plane.) Find the volume of the paperweight.

10. homework
Pg. 410 #1–6, 39–42