6-1 Estimating with finite sums

*Remember: Distance = Rate  Time *Finding distance traveled is simple with a constant rate! Ex) Think about a train moving 75 mph for 2 hours. What is the total distance traveled? Now represent this graph as an area. velocity 75 Area = 150 1 2 time Thing is, in real life this situation is very unlikely to happen! It would probably look more like this: Not easy to find area!

So how do we deal with this? We estimate areas using rectangles! The more the better! RAM: Rectangular Approximation Method LRAM uses the left endpoint for the height of the rectangle MRAM uses the midpoint for the height of the rectangle RRAM uses the right endpoint for the height of the rectangle

Try to find the area by dividing it up into strips (rectangles) Ex 1) A particle starts at x = 0 and moves along the x-axis with velocity v (t) = t2 for time t  0. Where is the particle at t = 3? Try to find the area by dividing it up into strips (rectangles) 3 Let’s take the above graph, from 0 to 3 and divide it into 6 subintervals. w = 0.5 LRAM A = (.5)(0)2 + (.5)(.5)2 + (.5)(1)2 + (.5)(1.5)2 + (.5)(2)2 + (.5)(2.5)2 = 6.875 0 .5 1 1.5 2 2.5 3

Let’s take the above graph, from 0 to 3 and divide it into 6 subintervals. w = 0.5 MRAM A = (.5)(.25)2 + (.5)(.75)2 + (.5)(1.25)2 + (.5)(1.75)2 + (.5)(2.25)2 + (.5)(2.75)2 = 8.9375 0 .5 1 1.5 2 2.5 3 RRAM A = (.5)(.5)2 + (.5)(1)2 + (.5)(1.5)2 + (.5)(2)2 + (.5)(2.5)2 + (.5)(3)2 = 11.375 0 .5 1 1.5 2 2.5 3

*To make calculating with more rectangles easier, we use the RAM program in our calculators. Enter the function in as y1; Exit PRGM; choose RAM Enter the specific values N? (# of subintervals) A? (left endpoint) B? (right endpoint) Let’s try what we just calculated by hand. N = 6, from 0 to 3 Now, try with 50 subintervals. And now 1000 subintervals. What do you notice about the results of the last RAM we ran? All 3 calculations are almost the same

This process of slicing and estimating can be used to find other areas and volumes. Ex 3) Estimate the volume of a sphere with radius 4. 4 –4 each little slice is a cylinder Calc: MRAM10 = 269.422986 MRAM100 = 268.0959772 Real value: rotate around x-axis

homework Pg. 274 # 1, 2, 5–7, 10, 12, 13, 25