Section 72a Area Between Curves Suppose we want to know the area of a region that is bounded above by one curve y f x and below by another y g x a b Upper curve ID: 613886
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Slide1
Area between curves
Section 7.2aSlide2
Area Between Curves
Suppose we want to know the area of a region that is bounded
above by one curve,
y = f(x), and below by another, y = g(x):
a
b
Upper curve
Lower curve
First, we partition the region
into vertical strips of equal
width and approximate
each strip as a rectangle
with area
Note: This expression will be
non-negative even if the region
lies below the
x
-axis.Slide3
Area Between Curves
Suppose we want to know the area of a region that is bounded
above by one curve,
y = f(x), and below by another, y = g(x):
a
b
Upper curve
Lower curve
We can approximate the
area of the region with the
Riemann sum
The limit of these sums as
isSlide4
Definition: Area Between Curves
If
f
and g are continuous with throughout[a, b], then the area between the curves y = f(
x) andy
= g(
x) from a
to b is the integral of [
f – g] from
a to b
,Slide5
Guided Practice
Find the area of the region between and
from to .
First, graph the two curves
over the given interval:
Now, use our new formula to
find the enclosed area:Slide6
Guided Practice
Find the area of the region between and
from to .
First, graph the two curves
over the given interval:Slide7
Guided Practice
Find the area of the region enclosed by the parabola
and the line .
The graph:
To find our limits of integration
(
a
and
b
), we need to solve
the system
a
b
Algebraically, or by calculator:Slide8
Guided Practice
Find the area of the region enclosed by the parabola
and the line .
The graph:
Because the parabola lies above
the line, we have
a
b
units
squaredSlide9
Guided Practice
Find the area of the region enclosed by the graphs of
and .
The graph:
a
b
To find our limits of integration
(
a
and
b
), we need to solve
the system
Solve graphically:
Store the negative value as A
and the positive value as B.Slide10
Guided Practice
Find the area of the region enclosed by the graphs of
and .
The graph:
a
b
Note: The trigonometric function
lies above the parabola…
units squared
Let’s evaluate this one numerically…
Area:Slide11
Guided Practice
Find the area of the region
R
in the first quadrant that is boundedabove by and below by the x-axis and the line .
The graph of
R
:
(4,2)
1
2
3
4
1
2
A
B
Area of region A:Slide12
Guided Practice
The graph of
R
:
(4,2)
1
2
3
4
1
2
A
B
Area of region B:Slide13
Guided Practice
The graph of
R
:
(4,2)
1
2
3
4
1
2
A
B
Area of
R
= Area of A + Area of B:
Units
squaredSlide14
Guided Practice
Find the area of the region enclosed by the given curves.
First, graph in [–2,12] by [0,3.5]
Intersection points:
Break into three
subregions
:Slide15
Guided Practice
Find the area of the region enclosed by the given curves.
First, graph in [–2,12] by [0,3.5]
Intersection points:
Break into three
subregions
: