AP Calculus 5.1 Areas and Distances

Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM)

Exercise 1: Calculate the area between the x-axis and the graph of 𝑦 = 3 − 2𝑥.

Exercise 2: Calculate the area between the x-axis and the graph of 𝑦 = √8 − 𝑥2

RECTANGULAR APPROXIMATION METHODS

Exercise 3: Use rectangles to estimate the area under the parabola 𝑦 = 𝑥2 from 0 to 1.

Solution 1: Using RRAM with 4 subintervals Solution 2: Using LRAM with 4 subintervals

Solution 3: Using MRAM with 4 subintervals

Exercise 4: Use rectangles to estimate the area under the curve 𝑦 = 1 + 𝑥3 from -2 to 3.

Solution 1: Using RRAM with 5 subintervals Solution 2: Using LRAM with 5 subintervals

Solution 3: Using MRAM with 5 subintervals

Exercise 5: Use rectangles to estimate the area under the curve 𝑦 = √𝑥 from 0 to 6.

Solution 1: Using RRAM with 12 subintervals Solution 2: Using LRAM with 12 subintervals

Solution 3: Using MRAM with 12 subintervals

Definition: The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the

areas of approximating rectangles:

𝐴 = lim𝑛→ ∞

𝑅𝑛 = lim𝑛→ ∞

[𝑓(𝑥1) ∆𝑥 + 𝑓(𝑥2) ∆𝑥 + 𝑓(𝑥3) ∆𝑥 + ⋯ + 𝑓(𝑥𝑛) ∆𝑥]

Using sigma notation

𝐴 = lim𝑛→ ∞

∑ 𝑓(𝑥𝑖)∆𝑥

𝑛

𝑖=1

Exercise 6: Let A be the area of the region that lies under the graph of 𝑓(𝑥) = cos 𝑥 between 𝑥 = 0 and 𝑥 = 𝜋

2.

Solution 1: Using RRAM with 4 subintervals

Solution 2: Using LRAM with 4 subintervals

Solution 3: Using MRAM with 4 subintervals

THE DISTANCE PROBLEM

Suppose you have the following problem to solve:

“A person drives their car at a rate of 50 mph for 3 hours straight. Find the distance traveled.”

Problems like the one above are fairly easy to solve, but do not model the real-world. What about the time where the

person had to get up to speed or slow down? If the velocity varies, it’s not as easy to find the distance traveled.

Example 5: Suppose the odometer in your car is broken and you want to estimate the distance driven over a 30 second

time interval. You take speedometer readings every 5 seconds and record them in the following table.

Time (s) 0 5 10 15 20 25 30

Velocity

(mph)

17 21 24 29 32 31 28

Draw a graph of the data.

Solution 1: Using RRAM

Solution 2: Using LRAM

How could you obtain a more accurate calculation without fixing your odometer?

AP Calculus 5.2 Riemann Sums and The Definite Integral

Goal: Calculate the exact area under a curve using Riemann Sums.

Exercise 1: Calculate the total distance traveled given the velocity equation. 𝑣(𝑡) = 𝑡2 − 2𝑡 − 8 𝑓𝑟𝑜𝑚 1 ≤ 𝑡 ≤ 6

Exercise 2: Calculate the total distance traveled given the acceleration equation. 𝑎(𝑡) = 𝑡 + 4

𝑣(0) = 5 𝑎𝑛𝑑 0 ≤ 𝑡 ≤ 10

Exercise 3: Calculate the total area under the curve of the 𝑓(𝑥) = 𝑥 [0,8]

Exercise 4: Calculate the total area under the curve of the 𝑓(𝑥) = 𝑥2 [0,2]

Exercise 5: Calculate the total area under the curve of the 𝑓(𝑥) = 3 + 𝑥2 [0,4]

AP Calculus 5.3 The Fundamental Theorem of Calculus

MENTAL OBJECTIVE: Discover & understand the relationship between net-area and antiderivatives.

PHYSICAL OBJECTIVE: Perform Riemann sum and integral calculations by hand and with a calculator.

PART 1: Let’s say you are walking at a certain speed. I’m not going to tell you the speed, because it’s changing, but I am

going to give you the position function:

Position function in feet: 𝑠(𝑡) = 𝑡2 + 5, 𝑡 is in seconds

Question: Find the displacement traveled during the first 5 seconds

Time Position (placement)

Start

End

PART 2: Compute the velocity function given 𝑠(𝑡) from PART 1. Complete the following chart to find the total

displacement during the first 5 seconds.

𝑡 0 1 2.5 3 4.75 5

𝑣(𝑡)

Calculate to distance using RRAM, LRAM, and find the exact by evaluating the integral.

Question: There is an easier, more unbelievable, seemingly improbable way to calculate the displacement rather than

finding the area under the velocity curve. How can you calculate the displacement, using an integral?

RRAM LRAM

EXACT

Displacement:

AP Calculus – Total Distance vs. Displacement Practice

The acceleration function in (𝑚/𝑠2) and the initial velocity are given for a particle moving along a line.

𝑎(𝑡) = 2𝑡 − 6, 𝑣(0) = 5, 0 ≤ 𝑡 ≤ 6

a) Calculate the displacement of the particle after the 5 seconds.

b) Calculate the distance traveled by the particle during the 5 seconds.

Given velocity or

acceleration

Function

𝒗(𝒕) = 𝟎 Position function Displacement

On Time Interval

Total Distance

On time interval

The Fundamental Theorem of Calculus, Part 1 (FTC 1)

The Fundamental Theorem of Calculus, Part 2 (FTC 2)

#1)

#2)

If 𝑓 is continuous on [𝑎, 𝑏], then

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)𝑏

𝑎

Where F Is any antiderivative of 𝑓, that is, a function 𝐹′ = 𝑓.

This tells you how to evaluate an integral.

Examples:

If 𝑓 is continuous on [𝑎, 𝑏], then the function 𝑔 defined by

𝑔(𝑥) = ∫ 𝑓(𝑡)𝑑𝑡 𝑎 ≤ 𝑥 ≤ 𝑏𝑥

𝑎

Is continuous on [𝑎, 𝑏] and differentiable on (𝑎, 𝑏), and 𝑔′(𝑥) = 𝑓(𝑥).

This says that the derivative and integral are opposite operations.

Examples:

Homework 5.3: Calculate the derivative of the function

𝑔(𝑥) 𝑔′(𝑥)

𝑔(𝑥) = ∫1

𝑡3 + 1𝑑𝑡

𝑥

1

𝑔(𝑥) = ∫ (2 + 𝑡4)5𝑑𝑡𝑥

1

𝑔(𝑥) = ∫ 𝑡2 sin 𝑡 𝑑𝑡𝑥

2

𝑔(𝑟) = ∫ √𝑥2 + 4 𝑑𝑥𝑟

0

𝑔(𝑥) = ∫ √1 + sec 𝑡 𝑑𝑡𝜋

𝑥

𝑔(𝑥) = ∫ sin3 𝑡 𝑑𝑡0

1𝑥2

𝑔(𝑥) = ∫ cos √𝑡 𝑑𝑡1

𝑥

𝑔(𝑥) = ∫ sin4 𝑡 𝑑𝑡

1𝑥

2

𝑔(𝑥) = ∫ √1 + 𝑟3𝑑𝑟𝑥2

0

𝑔(𝑥) = ∫ √𝑡 + √𝑡𝑑𝑡tan 𝑥

0

𝑔(𝑥) = ∫ (1 + 𝑣2)10𝑑𝑣cos 𝑥

1

HW: 5.3 Evaluate the integral with and without a calculator.

Integral

Work and answer without a calculator

Answer with a

calculator. Round to

3 decimals.

∫ (𝑥3 − 2𝑥)𝑑𝑥2

−1

∫ 6 𝑑𝑥5

−2

∫ (5 − 2𝑡 + 3𝑡2) 𝑑𝑡4

1

∫ (1 +1

2𝑥4 −

2

5𝑥9) 𝑑𝑥

1

0

∫ 𝑥45𝑑𝑥

1

0

∫ √𝑥3

𝑑𝑥8

1

∫3

𝑡4𝑑𝑡

2

1

∫ cos 𝜃 𝑑𝜃2𝜋

𝜋

∫ 𝑥(2 + 𝑥5) 𝑑𝑥2

0

∫ (3 + 𝑥√𝑥)𝑑𝑥1

0

∫𝑥 − 1

√𝑥𝑑𝑥

9

1

∫ (𝑦 − 1)(2𝑦 + 1)𝑑𝑦2

0

∫ sec2 𝑡 𝑑𝑡𝜋/4

0

AP Calculus 5.4 – Indefinite Integrals & Net Change

AP Calculus 5.5 – The “u”-Substitution Rule