1. Area under a curve
To the x axis

2. A=length*Width Upper vs. Lower Bound Sum of the area of the rectangles
Lower bound-inscribed
Sum of the area of the rectangles
Upper bound-circumscribed
A=length*Width
AP Tip-you can be tested on approximating area under a curve using rectangles and basic geo.

3. Midpoint
Use the midpoint value of f(x) for the height of the rectangle when calculating the area.

4. Some things to note
The smaller the rectangles, the better the area approximation will be. The problem will state how many rectangles you want to divide up into.
If the problem wants you to find an upper and lower bound to the area you need to give me that range, usually written as an inequality.
Yes you can use a calculator for these problems(if numbers are messy) I do expect you to know how to add fractions!

5. Definite Integral Guess what?
The definite integral gives us the exact area underneath the curve, so this is the best method to get our exact answer. Using rectangles gives us a good approximation. This is related to the Fundamental Theorem of Calculus Part 2

7. Calculator
The velocity of a high speed rail train is positive over 0 to 60 seconds. The velocity of the train is recorded below.
Estimate the acceleration of the train at t=25 seconds.
Use a Left Riemann sum with three subintervals of equal length to approximate
Find the average velocity over the 60 second period with a midpoint Riemann Sum(3 subintervals) t 10 20 30 40 50 60 v 45
105
140
165
195
210

8. Calculator
As a pot of coffee cools down, the temperature of the coffee is modeled by a differentiable function C, for [0,12]. Time is measured in minutes and temperature is measured in degrees Celsius. Use a trapezoid sum with 5 sub intervals to approximate
Interpret your answer
With units T 3 5 7 8 12 C 65 57 50 46 44 40

9. Calculator x 1 5 6 8 T
100 93 70 62 55
A metal wire of length 8 cm is heated at one end. The table above gives selected values of the temperature T(x), in degrees Celsius, of the wire x cm from the heated end. The function T is decreasing and twice differentiable.
Write an integral expression in terms of T(x) fro the average temperature of the wire. Estimate the average temperature using a trapezoidal sum with 4 sub intervals. Indicate the units of measure.’
Find interpret the meaning of your answer.

10. Fair game as possible test question next week