1. 4.2b Area Revisited

2. Smile! y = -x 2 +5 Find area from x=0 to 2 Width: Heights: f( ), f( ), f( ), . . . i th height? f( ) Area ≈ ∑ This would be called a lower sum, since it is an underestimate and all the rectangles are formed under the curve. It could also be called a right-hand sum, since all the rectangles are formed by the heights at the right hand side of the rectangle.

3. Width: Heights: f( ), f( ), f( ), . . . i th height? f( ) y = -x 2 +5 Find area from x=0 to 2 Area ≈ This would be called an upper sum, since it is an overestimate and all the rectangles are formed above the curve. It could also be called a left-hand sum, since all the rectangles are formed by the heights at the left hand side of the rectangle.

4. f(m i )=Minimum height for ith interval f(M i )=Maximum height for ith interval Refer to p. 263

5. Refer to p. 263

6. Find the upper and lower sums for the region bounded by the graph of f(x) = x 2 and the x-axis between x = 0 and x = 2 Since f is increasing on interval, lower sum rectangles form from the left endpoint of each interval. f(x) = x 2 Upper sum rectangles form from the right endpoint of each interval Ex 4 p. 264

7. As n increases, these two sums get closer to the same value. Refer to p. 265

8. Refer to p. 265

9. Ex 5 p. 266 Finding area by using the limit definition Find area under graph f(x) = x 3 , above the x-axis, and between x=0 and x = 1

10. Ex 6 p. 266 Finding area by using the limit definition Find area under graph f(x) = 4 – x 2 , above the x-axis, and between x=1 and x = 2

11. Ex 7 p. 267 A region bounded by y-axis Find the area of the region bounded by the graph of f(y) = y 2 , the y-axis, and 0 ≤ y ≤ 1 When f is a continuous, nonnegative function of y, you can still use same techniques.

12. http://www.math.psu.edu/dlittle/java/calculus/area.html 4.2b p. 267/ 23-29 odd, 33-42 mult of 3, 49, 53, 59 A great resource for visualizing and checking answers is:

13. http:// youtu.be/ZXKowQRwuwA for left hand sums