Drill: Evaluate each sum

1. Drill: Evaluate each sum • Recall that • 12 + 22 + 32 + 42 + 52 =55 • [3(0) -2] + [3(1) -2] + [3(2) -2] + [3(3) -2] + [3(4) -2] = 20 • 100 (0 + 1)2 + 100 (1 + 1)2+ 100 (2 + 1)2+ (3 + 1)2+ 100 (4 + 1)2=5500

2. Write the sum in sigma notation • 1 + 2 + 3 + …..+ 98 + 99 • 0 + 2 + 4 + ….48 + 50 • 3(1)2 + 3(2)2 + …. 3(500)2

3. Definite Integrals Lesson 5.2

4. Objectives • Students will be able to • express the area under a curve as a definite integral and as a limit of Riemann sums. • compute the area under a curve using a numerical integration procedure.

5. Key Concept: Riemann Sum A Riemann sum, Rn, for function f on the interval [a, b] is a sum of the form where the interval [a, b] is partitioned into n subintervals of widths Δxk, and the numbers {ck} are sample points, one in each subinterval.

6. Example: Calculating Riemann Sums Upper = using right endpoints: ¼ ( 1/8 + 27/64 + 1) = 99/256 Lower: using left endpoints: ¼ ( 1/64 + 1/8 + 27/64) = 9/64

7. Definite Integral The function is the integrand Upper limit of integration Let f be continuous on [a,b] and be partitioned into n subintervals of equal length Δx = (b – a)/n ck is some point in the kth subinterval. When you find the value of the integral, you have evaluated the integral Integral sign x is the variable of integration lower limit of integration

8. Express each limit as a definite interval

9. The Integral of a Constant

10. Key Concept: Area Under a Curve If y = f (x) is nonnegative and integrable on [a, b], and if Rn is any Riemann sum for f on [a, b], then

11. Use the graph of the integrand and areas to evaluate the integral 5 2 6 A=1/2(6)(2+5)= 21

12. Homework • Day 1: p. 282/3: 3-6, 8-12, 14, 29-30 • Day 2: p. 283: 16-22, 33-36 (NINT means fnInt on your calculator)

13. Drill (let h be measured in feet) h(t) = -5t2 + 20t + 15 • a. Estimate the instantaneous velocity of the ball 3 seconds after it’s thrown. • b. Estimate the acceleration of the ball 3 seconds after it’s thrown. • c. Estimate the maximum height. • v(t) = -10t + 20 • v(3) = -30 + 20 = -10 ft/s • a(t) = -10 ft/ s2 • 0 = -10t + 20 • t = 2 • h(2) = 35 feet

14. Example Area Under a Curve Determine the area under the curve over the interval [–4, 4].

15. Example Area Under a Curve Evaluate the integral.

16. Example: Estimate • Solution: Graph the function. It is a semi-circle. The shaded area represents the area of the region bounded by the semicircle, the x-axis, and the lines x =2 and x = 3. • Determine the area of the sector of the circle: ½r2θ, where r is radius and θ is the angle in RADIANS! • You will need to find θ by tan-1(y/x) • Determine the area of the triangle: ½bh or ½ xy • Subtract the two areas.

17. Θ 3 (x,y) = (2, ) = (2, ) • r = 3 • Θ= tan-1( /2)= • .84 radians • ½r2θ = 3.78 • A = ½ (2) = 2.24 • 3.78-2.24 = 1.54 Θ Θ 2 2 3

18. Using your calculator to determine integrals • Math • 9: fnInt • fnInt(f(x), x , a, b)

19. Example Using Your Calculator to Evaluate Integrals Approximate the following integrals using a calculator.