AP Calculus AB
Chapter 4, Section 1
Integration
2013 - 2014
Derivatives vs. Integrals
• We use derivatives to find slopes of tangent lines or the
___________________ of a function at a specific point.
• We use integrals to find the
_____________________________.
• Integrals are often called ________________.
Derivatives vs. Integrals
• How are the three given functions related?
• 𝑓 𝑥 = 𝑥 3 + 2𝑥 2 + 7𝑥 − 10
• 𝑔 𝑥 = 3𝑥 2 + 4𝑥 + 7
• ℎ 𝑥 = 6𝑥 + 4
Integrals
• Can you find function F whose derivative is 𝑓 𝑥 = 3𝑥 2 ?
Is your answer the only possible answer?
General Solution
• Remember that a derivative will not include any constants
from a given function. So account for the constant that may
or may not be present in the original function, we add C to the
end of the antiderivative.
Notation for Antiderivatives
𝑑𝑦
• When solving a differential equation in the form = 𝑓(𝑥), it is
𝑑𝑥
convenient to write the equation in differential form:
• The solutions to the antiderivative (or indefinite integration) is
denoted by the integral sign ∫.
𝑦=
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶
Something to remember
• Derivatives and Integrals are basically “inverses” of each other.
• Basic Integration Rules on page 250
Applying the Basic Integration
Rules
• Describe the antiderivatives of 3x.
Rewriting Before Integrating
Original Integral
1
𝑑𝑥
𝑥3
𝑥 𝑑𝑥
2 sin 𝑥 𝑑𝑥
Rewrite
Integrate
Simplify
Integrating Polynomial
Functions
•
𝑑𝑥
Integrating Polynomial
Functions
•
(𝑥 + 2)𝑑𝑥
Integrating Polynomial
Functions
•
(3𝑥 4 − 5𝑥 2 + 𝑥)𝑑𝑥
Rewriting Before Integrating
•
𝑥+1
𝑑𝑥
𝑥
Rewrite Before Integrating
•
sin 𝑥
𝑑𝑥
cos2 x
Initial Conditions and Particular
Solutions
• The type of antiderivatives you have been learning about are
all vertical translations of each other (due to the addition of
C).
• In many situations, you are given enough information to find
the particular solution. To determine the value of C, all you
need to know is __________________________.
Finding a Particular Solution
• Find the general solution of
𝐹′ 𝑥 =
1
,
𝑥2
𝑥>0
and find the particular solution that satisfies the initial condition F(1) =
0.
Solve the differential equation
• 𝑓 ′′ 𝑥 = 2, 𝑓 ′ 2 = 5, 𝑓 2 = 10
Solving a Vertical Motion
Problem
• A ball is thrown upward with an initial velocity of 64 feet per
second from an initial height of 80 feet.
• Find the position of the function giving the height x as a
function of time t.
• When does the ball hit the ground?
(Use -32 feet per second as the acceleration due to gravity.)
Ch. 4.1 Homework
• Pg 255 – 257, #’s: 1, 7, 9, 13, 15, 23, 29, 35, 39, 45, 55, 61, 69
• 13 total problems