1. 14. 2 Double Integration For more information visit http://www. math

2. Example 1
Approximate the volume of the solid lying between the paraboloid f(x,y) and the square region R given by the plane 0 < x < 1 and 0 < y < 1 use a partion made up of squares having length of side ¼
(this problem is similar to MRAM from Calc AB)

3. Solution to Ex 1
It is convenient to choose the centers of the subregions as the points to evaluate f(x,y). Each sub region forms a rectangle with length ¼ and width ¼ and the height is the value of the function. The following values of are the centers of the sub regions

4. Solution to Example 1 cont
The volume can be approximated by

6. Find the volume under this surface between 0<x<2 and 0<y<1.

7. z
We can sketch the graph by putting in the corners where (x=0, y=0), (x=2, y=0),
(x=0, y=1), (x=2, y=1). y x

8. We could hold x constant and take a slice through the shape.y z
The area of the slice is given by:
The volume of the slice is area . thickness

9. We can add up the volumes of the slices by:x y z

10. The base does not have to be a rectangle:
with triangular base between the x-axis, x=1 and y=x. x y
thickness of slice
area of slice
slice
Add all slices from 0 to 1.

13. Example 2

14. Solution to Example 2

16. Homework p. 997 1,3,13-23 odd (do not need to do approx for problem 3)
No human investigation can be called real science if it cannot be demonstrated mathematically.
-- Leonardo da Vinci
Teacher: How much is half of 8?
Pupil: Up and down or across?
Teacher: What do you mean?
Pupil: Well, up and down makes a 3
or across the middle leaves a 0!

17. Figure 14.17

18. Figure 14.16

19. Figure 14.15