2. 5.1 Estimating with Finite Sums Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 Greenfield Village, Michigan

3. time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3 ft/sec. Since rate. time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

4. If the velocity is not constant, we might guess that the distance traveled is still equal to the area under the curve. (The units work out.) Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. This is called the Left-hand Rectangular Approximation Method (LRAM). Approximate area: (too low)

5. We could also use a Right-hand Rectangular Approximation Method (RRAM). Approximate area: (too high)

6. Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). Approximate area: In this example there are four subintervals. As the number of subintervals increases, so does the accuracy. (much closer)

7. Approximate area: width of subinterval With 8 subintervals: The exact answer for this problem is.

8. Averaging right and left rectangles gives us trapezoids:

9. (too high)

10. Trapezoidal Rule: ( h = width of subinterval ) Or you could just average LRAM and RRAM – easier than memorizing the formula!

11. Compare this with the Midpoint Rule: Approximate area: (too low)0.625% error The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

12. Midpoint Rule: (too low)0.625% error Trapezoidal Rule: 1.25% error (too high) Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction. If we use a weighted average: This is the exact answer! Oooh! Ahhh! Wow!

13. Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve: Riemann Sums

14. Practice: Example 1: Compute LRAM, MRAM and RRAM for from x = 0 to x = 3 using 3 subintervals. LRAM ≈ 2.414 MRAM ≈ 3.513 RRAM ≈ 4.146

15. Practice: Example 2: Approximate the area under the curve of y = -x 2 + 5 from x = 0 to x = 2 using 4 subintervals and the trapezoidal method. 7.25

16. Practice: Example 3: Planes have instruments that measure their velocity. Suppose a jet takes off, becomes airborne at a velocity of 180 mph and climbs to cruising altitude. Using the table of velocities, approximate how far the jet has flown. Time (min)012457 Velocity (mph)180240300420480530 Between 40 and 48.667 miles

17. We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation. The TI-89 calculator can do these rectangular approximation problems. This is of limited usefulness, since we will learn better methods of finding the area under a curve, but you could use the calculator to check your work.

18. If you have the calculus tools program installed: Set up the WINDOW screen as follows:

19. Select Calculus Tools and press Enter Press APPS Press F3 Press alpha and then enter: Make the Lower bound: 0 Make the Upper bound: 4 Make the Number of intervals: 4 Press Enter and then 1 Note: We press alpha because the screen starts in alpha lock.

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