1. 4-2 AREA AP CALCULUS – MS. BATTAGLIA

2. SIGMA NOTATION The sum of n terms a 1, a 2, a 3,…, a n is written as where i is the index of summation, a i is the ith term of the sum, and the upper and lower bounds of summation are n and 1.

3. EXAMPLES OF SIGMA NOTATION

5. SUMMATION FORMULAS

6. THM 4.2 SUMMATION FORMULAS Evaluate

7. APPROXIMATING THE AREA OF A PLANE REGION Use left and right endpoints and 5 rectangles to find two approximations of the area of the region between f(x)=-x 2 + 5 and the x-axis over the interval [0,2].

8. FINDING UPPER AND LOWER SUMS FOR A REGION 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.

9. THM 4.3 LIMITS OF THE LOWER AND UPPER SUMS Let f be continuous and nonnegative on the interval [a,b]. The limits as n   ∞ of both the lower and upper sums exist and are equal to each other. That is, where Δx=(b - a)/n and f(m i ) and f(M i ) are the minimum and maximum values of f on the subinterval.

10. FINDING AREA Find the area of the region bounded by the graph of f(x) = x 3, the x-axis, and the vertical lines x=0 and x=1.

11. FINDING AREA Find the area of the region bounded by the graph of f(x)=4-x 2, the x-axis, and the vertical lines x = 1 and x = 2.

12. HOMEWORK P. 268-269 #27-30, 57-60