areas between curves...in calculus i we learned how to find the area under a curve bounded by the...
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6.1– Areas Between Curves
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al_calculus/area_between_two_curves/area_between_two_curves.html
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In Calculus I we learned how to find the area under a
curve bounded by the x-axis.
In this section we are going to learn how to find areas
between curves.
Example →
3
As we did for areas under curves in Section 5.1, we
divide S into n strips of equal width and approximate
the i th strip by a rectangle with base ∆x and height
( *) ( *)i if x g x
AREAS BETWEEN CURVES
Consider area bounded by the functions below.
The width of the rectangle will be Dx and
Since that change will be
very small we will call it dx,
and the height will be
5
2( ) 5f x x
( ) 2g x x
1 2
2
2
( ) ( )
5 ( 2)
7
f x g x y y
x x
x x
1y
2y
http://www.slu.edu/classes/maymk/banchoff/AreaBetweenCurves.html
Then the area of the rectangle will be
So the area between
the curves will be the
sum of all rectangles
That is
a, and b are ????
6
2( ) 5f x x
( ) 2g x x
2 7length width x x dx
1y
2y 2 7
b
a
x x dx
Formula for the area between curves:
6.1 Areas Between Curves 7
The area of the region bounded by
the curves y=f(x) and y=g(x) and
the lines x=a and x=b where f and
g are continuous and f(x) g(x)
for all x in [a,b] is given by
Areab
af x g x dx
y = f(x)
y = g(x)
We can find the area
between the two
functions if we divide
it in two pieces
So the total area would
be
Sometimes we will have to find areas that look like the
one below ▼
8
2y x
y x y x
2y x A1
A2
1 2
2 4
0 2 2
A A
x dx x x dx
9
There is however an easier method;
Instead of using Vertical Strips we could use
Horizontal Strips, which would give us
22
02 y y dy
y x
2y x
dy
y x
2y x
2y x
2y x
2
2 3
0
1 12
2 3y y y
82 4
3
10
3
10
1
Decide on vertical or horizontal strips. (Pick whichever is easier to write
formulas for the length of the strip, and/or whichever will let you
integrate fewer times.)
Sketch the curves.
2
3 Write an expression for the area of the strip.
(If the width is dx, the length must be in terms of x.
If the width is dy, the length must be in terms of y.
4 Find the limits of integration. (If using dx, the limits are x values; if
using dy, the limits are y values.)
5 Integrate to find area.
11
Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to x or y. Find the area
of the region.
2
2
1. 1, 9 , 1, 2
2. sin , , 0, / 2
3. tan , 2sin , / 3 / 3
4. 4 12,
5. cos , 1 cos , 0
x
y x y x x x
y x y e x x
y x y x x
x y x y
y x y x x
12
Evaluate the integral and interpret it as the area of
a region. Sketch the region.
Solutions
/2
0
4
0
6. | sin cos 2 |
7. | 2 |
x x dx
x x dx
13
Use a graph to find the x-coordinates of the points of
intersection of the given curves. Then find (possibly
approximately) the area of the region bounded by
the curves.
2 4
2
2 3
8. sin( ),
9. , 2
10. 3 2 , 3 4
x
y x x y x
y e y x
y x x y x x
http://archives.math.utk.edu/visual.calculus/5/area2curves.3/index.html
http://www.math.dartmouth.edu/~klbooksite/4.07/407.html
http://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx
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