This document discusses calculating area under a curve using left and right Riemann sums. It explains that lower/right sums underestimate the area, using heights on the right side of each interval, while upper/left sums overestimate the area using heights on the left side. Examples are provided to calculate these sums and find the area under graphs between given bounds as the number of intervals increases. Additional examples demonstrate finding areas when the curve is defined in terms of y instead of x.

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