Transcript 7 1 Area Btw Curves

7-1: Area Between Curves
Objectives:
1. To find the area
between two curves
Assignment:
• P. 452-454: 1-5 odd, 15,
16, 17-29 eoo, 35, 39,
43, 47, 49, 52, 71, 75,
88, 90-92, 94
Warm Up 1
Where does the graph
of 𝑓 𝑥 = 2𝑥 intersect
the graph of
𝑔 𝑥 = 𝑥 2 − 4𝑥 + 5?
Warm Up 2
What is the length
of 𝐴𝐵?
Warm Up 3
Where does the graph of
𝑓 𝑥 = 2𝑥 intersect the graph
of 𝑔 𝑥 = 𝑥 2 − 4𝑥?
Warm Up 4
What is the length of 𝐴𝐵?
Objective 1
You will be able to find the area
between two curves
Area Between Curves
Let’s say that
instead of finding
the area under a
curve, we want to
find the area
between two
curves.
Area Between Curves
We could
approximate
the area by
subdividing
the interval
into 𝑛
rectangles
with width ∆𝑥.
Width = ∆𝑥
Area Between Curves
The height of
one of these
rectangles is
the difference
in the 𝑦-values
evaluated at
some 𝑥-value
in each
subinterval.
Representative Rectangle
Width = ∆𝑥
Height = 𝑓 𝑥𝑖∗ − 𝑔 𝑥𝑖∗
Top
Bottom
Area Between Curves
The area
could be
approximated
by a Riemann
sum.
Width
𝑛
𝑓 𝑥𝑖∗ − 𝑔 𝑥𝑖∗
𝐴≈
𝑖=1
How could we
get a better
approximation?
Height
∆𝑥
Area Between Curves
As 𝑛 → ∞, the
Riemann sum
approaches the
actual area
between the two
curves.
𝑛
𝑓 𝑥𝑖∗ − 𝑔 𝑥𝑖∗ ∆𝑥
𝐴 = lim
𝑛→∞
𝑖=1
𝑏
𝐴=
𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥
𝑎
Area Between Curves
If 𝑓 and 𝑔 are continuous
on 𝑎, 𝑏 and 𝑔 𝑥 ≤ 𝑓 𝑥
for all 𝑥 in 𝑎, 𝑏 , then the
area of the region
bounded by the graphs of
𝑓 and 𝑔 and the vertical
lines 𝑥 = 𝑎 and 𝑥 = 𝑏 is
𝑏
𝐴=
𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥
𝑎
Area Between Curves
Intuitively, this should make sense, since…
Representative Rectangle
For each of these problems, use a representative
rectangle to write a Riemann sum, the limit of
which is a definite integral.
Representative Rectangle
Riemann Sum
Definite Integral
Representative Rectangle
For each of these problems, use a representative
rectangle to write a Riemann sum, the limit of
which is a definite integral.
Representative Rectangle
Riemann Sum
Definite Integral
Representative Rectangle
For each of these problems, use a representative
rectangle to write a Riemann sum, the limit of
which is a definite integral.
Representative Rectangle
This will help
identify the top and
bottom curves
Riemann Sum
Definite Integral
It will also help later
when setting up
volume problems
Exercise 1
Find the area of the
region bounded by the
graphs of 𝑦 = 𝑥 2 + 2,
𝑦 = −𝑥, 𝑥 = 0, and
𝑥 = 1.
Exercise 2
Find the area of the
region bounded by
the graphs of 𝑦 = 𝑒 𝑥 ,
𝑦 = 𝑥, 𝑥 = 0, and
𝑥 = 1.
Intersecting Graphs
Occasionally, we want
to find the area
bounded by two
intersecting graphs.
In this case, to find the
limits of integration,
we must first find the
points of intersection
by solving the system
of equations.
Exercise 3
Find the area of
the region bound
by the graphs of
𝑓 𝑥 = 2 − 𝑥 2 and
𝑔 𝑥 = 𝑥.
Exercise 4
The sine and cosine
graphs intersect an
infinite number of
times, bounding
regions of equal
area. Find the area
of one of these
regions.
Exercise 5
Find the area of the region
between the graphs of
𝑓 𝑥 = 3𝑥 3 − 𝑥 2 − 10𝑥
and 𝑔 𝑥 = −𝑥 2 + 2𝑥.
Exercise 6: AP FRQ
Let 𝑓 and 𝑔 be functions
defined by
2 −2𝑥
𝑥
𝑓 𝑥 =1+𝑥+𝑒
and
𝑔 𝑥 = 𝑥 4 − 6.5𝑥 2 + 6𝑥 + 2.
Let 𝑅 and 𝑆 be the two
regions enclosed by the
graphs of 𝑓 and 𝑔 shown in
the figure. Find the sum of
the areas of regions 𝑅 and
𝑆.
Horizontal Rectangles
All of the preceding problems used a vertical
representative rectangle. Sometimes, however, it’s
best to use a horizontal rectangle and to integrate
with respect to 𝑦.
Exercise 7
Find the area
bounded by the
graphs of 𝑥 = 3 − 𝑦 2
and 𝑥 = 𝑦 + 1.
Exercise 7 (Revisited)
Find the area
bounded by the
graphs of 𝑥 = 3 − 𝑦 2
and 𝑥 = 𝑦 + 1.
Exercise 7 (Revisited)
Vertical vs. Horizontal
Functions of 𝑥
Height = top − bottom
Functions of 𝑦
Height = right − left
𝑏
𝐴=
𝑑
𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥
𝑎
𝐴=
𝑓 𝑦 − 𝑔 𝑦 𝑑𝑦
𝑐
How exactly
will you know
when to use
vertical versus
horizontal
rectangles?
7-1: Area Between Curves
Objectives:
1. To find the area
between two curves
Assignment:
• P. 452-454: 1-5 odd, 15,
16, 17-29 eoo, 35, 39,
43, 47, 49, 52, 71, 75,
88, 90-92, 94