Over Lesson 12-5
A garden is in the shape of the region shown below.
If each unit in the coordinate system represents one
foot, approximately how much area does the garden
contain?
A. 17.25 ft2
B. 22 ft2
C. 26.25 ft2
D. 28.75 ft2
Over Lesson 12-5
A garden is in the shape of the region shown below.
If each unit in the coordinate system represents one
foot, approximately how much area does the garden
contain?
A. 17.25 ft2
B. 22 ft2
C. 26.25 ft2
D. 28.75 ft2
You used limits to approximate the area under a curve.
(Lesson 12-5)
• Find antiderivatives.
• Use the Fundamental Theorem of Calculus.
• antiderivative
• indefinite integral
• Fundamental Theorem of Calculus
Find Antiderivatives
A. Find an antiderivative for the function f(x) = 6x.
We need to find a function that has a derivative of 6x.
Recall that the derivative has an exponent that is one
less than the exponent in the original function.
Therefore, F(x) will be raised to a power of two. Also,
the coefficient of a derivative is determined in part by
the exponent of the original function. F(x) = 3x2 fits this
description. The derivative of 3x2 is 2(3)x2 – 1 or 6x.
However, 3x2 is not the only function that works. The
function G(x) = 3x2 + 5 is another because the
derivative is G′(x) = 2(3)x2 – 1 + 0 or 6x. Another
answer would be H(x) = 3x2 – 12.
Answer: Sample answer: 3x2
Find Antiderivatives
B. Find an antiderivative for the function f(x) = –6x–7.
Again, the derivative has an exponent that is one less
than the exponent in the original function, so F(x) will
be raised to the negative sixth power. We can try
F(x) = x–6. The derivative of x–6 is –6x–6 –1 or –6x–7.
G(x) = x–6 + 4 and H(x) = x–6 – 8 are other
antiderivatives.
Answer: Sample answer: x–6
Find an antiderivative of
A. F(x) = x–6
B.
C.
D. F(x) = –4x–6 –2
.
Antiderivative Rules
A. Find all antiderivatives for the function f(x) = 3x5.
Original equation
Constant Multiple of
a Power
Simplify.
Answer:
Antiderivative Rules
B. Find all antiderivatives for the function.
Original equation
Rewrite with a negative
exponent.
Constant Multiple of a
Power
Simplify.
Answer:
Antiderivative Rules
C. Find all antiderivatives for the function
f(x) = x2 + 3x + 4.
Original equation
Rewrite the function
so each term has a
power of x.
Antiderivative Rule
Simplify.
Answer:
Find all antiderivates for
A.
B.
C.
D.
.
Indefinite Integral
A. CLIFF DIVER A cliff diver is diving off a cliff
that is 100 feet high. His instantaneous velocity
can be defined as v(t) = –32t, where t is given in
seconds and velocity is measured in feet per
second. Find the position function s(t) of the diver.
To find the function for the position of the diver, find the
antiderivative of v(t).
dt
dt
Relationship between
position and velocity
v(t) = –32t
Indefinite Integral
Constant Multiple of a
Power
Simplify.
Find C by substituting 100 feet for the initial height and
0 for the initial time.
Antiderivative of v(t)
s(t) = 100 and t = 0
Simplify.
Indefinite Integral
The position function for the diver is s(t) = –32t 2 + 100.
Answer: s(t) = –32t 2 + 100
Indefinite Integral
B. CLIFF DIVER A cliff diver is diving off a cliff
that is 100 feet high. His instantaneous velocity
can be defined as v(t) = –32t, where t is given in
seconds and velocity is measured in feet per
second. Find how long it will take for the diver to
reach the water.
Solve for t when s(t) = 0.
Position function for the
diver
s(t) = 0
Subtract 100 from each
side.
Indefinite Integral
Divide each side by –16.
Take the positive square
root of each side.
The diver will reach the water in 2.5 seconds.
Answer: 2.5 s
SKY DIVING A sky diver is jumping from an airplane
that is 5776 feet above the height she intends to pull
the chord on her parachute. Her instantaneous
velocity can be defined as v(t) = –32t, where t is
given in feet per second. Find the position function
s(t) of the sky diver relative to where she wants to
pull the chord. How long does it take the sky diver
to reach the point where she will pull the chord?
A. v(t) = –16t 2 + 5776; t = 19 s
B. v(t) = –16t 2 + 5776; t = 361 s
C. v(t) = –16t 2 + 361; t = 19 s
D. v(t) = 015032t 2 + 5776; about 10.4 s
Area Under a Curve
A. Use the Fundamental Theorem of Calculus to
find the area of the region between the graph of
the function y = 5x4 and the x-axis on the interval
[2, 4], or
.
First, find the antiderivative.
Constant Multiple
of a Power
Simplify.
Area Under a Curve
Now evaluate the antiderivative at the upper and lower
limits and find the difference.
Fundamental Theorem of
Calculus
b = 4 and a = 2
Simplify.
The area between the graph and the x-axis on the
interval [2, 4] is 992 square units.
Answer: 992 units2
Area Under a Curve
B. Use the Fundamental Theorem of Calculus to
find the area of the region between the graph of
the function y = –x2 + 6x + 9 and the x-axis on the
interval [0, 6] or
.
First, find the antiderivative.
Antiderivative Rule
Area Under a Curve
Simplify.
Area Under a Curve
Now evaluate the antiderivative at the upper and lower
limits and find the difference.
Fundamental
Theorem of
Calculus
b = 6 and a = 0
Area Under a Curve
Simplify.
The area between the graph and the x-axis on the
interval [0, 6] is 90 square units.
Answer: 90 units2
Use the Fundamental Theorem of Calculus to find
the area of the region between the graph of
y = x4 + 4 and the x-axis on the interval [0, 2].
A. 0.8 units2
B. 14.4 units2
C. 5.8 units2
D. 6.4 units2
Indefinite and Definite Integrals
A. Evaluate
dx.
This is an indefinite integral. Use the antiderivative
rules to evaluate.
Constant
Multiple of a
Power
Simplify.
Answer:
Indefinite and Definite Integrals
B. Evaluate
dx.
This is a definite integral. Evaluate the integral using
the given upper and lower limits.
Fundamental
Theorem of
Calculus
b = 4 and a = 1
Indefinite and Definite Integrals
Simplify.
The area between the graph and the x-axis on the
interval [1, 4] is 51.75 square units.
Answer: 51.75
Evaluate
A.
B.
C.
D.
.
Definite Integrals
The work, in joules, required to stretch a certain
spring is given by
. How much work is
required?
Evaluate the definite integral for the given upper and
lower limits.
Constant Multiple
of a Power and the
Fundamental
Theorem of
Calculus
Definite Integrals
Let a = 0 and
b = 2.5 and subtract.
Simplify.
The work required is 187.5 joules.
Answer: 187.5 J
The work, in joules, required to stretch a certain
spring is given by
required?
A. 60 J
B. 180 J
C. 270 J
D. 360 J
. How much work is
• antiderivative
• indefinite integral
• Fundamental Theorem of Calculus