1. 4.3 Day 1 – Day 1 Review
Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening and is recorded in the table.
Estimate the total quantity of oil that has escaped after 5 hours using a trapezoidal sum
Time (h) 1 2 3 4 5
Leakage (gal/h) 50 70 97
136
190
265

2. Section 4.3 Day 3 Riemann Sums & Definite Integrals
AP Calculus AB

3. Learning Targets Define Riemann Sums
Conceptually connect approximation and limits
Evaluate left hand, right hand and midpoint Riemann Sums of equal and unequal lengths from graphs & tables
Evaluate approximations using the trapezoidal rule
Define a definite integral
Evaluate a definite integral geometrically and with a calculator
Define an integral in terms of area
Apply properties of a definite integral

4. Definite Integral Definition
𝑎 𝑏 𝑓 𝑥 𝑑𝑥
Representation for the exact area under the curve
Interval: [𝑎, 𝑏]
Graph in consideration: 𝑓(𝑥)
𝑥 is the variable of integration
Note: you can use any letter in place of 𝑥

5. Definite Integral Connection to Riemann Sum
lim 𝑛→∞ 𝑘=1 𝑛 𝑓 𝑐 𝑘 ∆𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
The limit/summation becomes the integral sign, 𝑓 𝑐 𝑘 =𝑓(𝑥) and ∆𝑥=𝑑𝑥
Note: we call 𝑑𝑥 a differential

6. Definite Integral Interpretation
Example 1: Find 0 4 2𝑥 𝑑𝑥 Interval: [0,4] Function: 𝑓 𝑥 =2𝑥 Integral implies area under the curve Find the area of the triangle

7. Definite Integral Interpretation
Example 2: Find −2 5 3 𝑑𝑥 Interval: [−2, 5] Function: 𝑓 𝑥 =3 Integral implies area under the curve Find the area of the rectangle

8. Calculator Functions
Integration: -Click Math, then scroll down to “fnint” fnint (expression, variable of integration, lower bound, upper bound) Derivative at a Point: -Click Math, then scroll down to “nderiv” nderiv(expression, variable, value)

9. Calculator Functions
Graphing Features: Graph 𝑦= 𝑥 2 Then, go to 2nd Trace/Calc Click 𝑑𝑦/𝑑𝑥 to find the derivative at a point Click 𝑓(𝑥) 𝑑𝑥, enter the lower and upper limit, and calculate the area under the curve

10. Definite Integral Interpretation
Example 3: Sketch a graph and shade the region of the following functions. Then, use “fnint” to find the value of the definite integrals 𝑥 2 +1 𝑑𝑥 2. −2 0 𝑥 2 +1 𝑑𝑥 3. 0 −2 𝑥 2 +1 𝑑𝑥 𝑥 2 −1 𝑑𝑥 5. −2 2 sin⁡(𝑥) 𝑑𝑥

11. Definite Integral Interpretation
𝑥 2 +1 𝑑𝑥 −2 0 𝑥 2 +1 𝑑𝑥 −2 𝑥 2 +1 𝑑𝑥
𝑥 2 −1 𝑑𝑥 −2 2 sin⁡(𝑥) 𝑑𝑥
Answer the questions based on your answers from before.
1. Why is the answer to #3 a negative number?
2. Why is the answer to #4 a negative number?
3. Explain the answer for #5.

12. Definite Integral Properties
1. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥=0 2. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 3. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥+ 𝑐 𝑏 𝑓(𝑥) 𝑑𝑥 4. 𝑎 𝑏 𝑘𝑓(𝑥) 𝑑𝑥=𝑘 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 5. 𝑎 𝑏 [𝑓 𝑥 ±𝑔 𝑥 ] 𝑑𝑥= 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥± 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥 6. If a function is continuous on [𝑎,𝑏], then it is integrable on [𝑎,𝑏].

13. Definite Integral Properties
Example 1: Evaluate 𝟏 𝟑 − 𝒙 𝟐 +𝟒𝒙−𝟑 𝒅𝒙 given 𝟏 𝟑 𝒙 𝟐 𝒅𝒙= 𝟐𝟔 𝟑 , 𝟏 𝟑 𝒙 𝒅𝒙=𝟒, 𝟏 𝟑 𝒅𝒙 =𝟐 = 𝟒 𝟑

14. Definite Integral Properties
Example 2: Evaluate 𝟐 𝟓 𝒇 𝒙 +𝟒 𝒅𝒙 , given 𝟐 𝟓 𝒇 𝒙 𝒅𝒙 =𝟏𝟖 𝟐 𝟓 𝒇 𝒙 𝒅𝒙 + 𝟐 𝟓 𝟒𝒅𝒙 =𝟏𝟖+𝟏𝟐=𝟑𝟎

15. Train Problem Revisited
A train moves along a track at a steady 𝟕𝟓 miles per hour from 𝟕:𝟎𝟎 am to 𝟗:𝟎𝟎 am. Express its total distance traveled as an integral. Distance traveled: 𝟕 𝟗 𝟕𝟓 𝒅𝒕 =𝟕𝟓∙ 𝟗−𝟕 =𝟏𝟓𝟎