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8/3/2019 Area Between 2 Curves
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The Area Between 2 Curves
Calculus, Section 6.1
State Standard: Calculus 16.0
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Concept of a Region
A region is the area/space trapped between 2 intersecting
curves or the area/space between 2 curves and 2 vertical lines
that act as boundaries.
Upper
curve
(top)
Lower
curve
(bottom
Region
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Concept of a Region (2)
Sometimes the curves
take turns being on the
top and the bottom
Sometimes the curve is the
top and bottom at the same
time.
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Area of a Region1. Divide the region into a series
of rectangles.
2. Find the area of each rectangle.
3. Add the areas together to getan approximation of the total
area.
4. The greater the number of
rectangles, the less error.
5. An infinite number of
rectangles will completely
squeeze out the error.
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Area of a Rectangle
The area of the rectangle is its
height times its width.
We will call the width
The height is the distance
(difference) between the upper
curve and the lower curve. Think
of it as the topthe bottom.
The area becomes
x
( ) ( )f x g x x
f(x)
( ) ( )f x g x
x
g(x)
(a, f(a))
(b, f(b))
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Area of the Region
To get the area of the region, we
find the area of each rectangle
and add them up.
1
( ) ( )n
i i i
i
f x g x x
To get rid of the error, we
take an infinite number of
rectangles squeeze the error
out.
1
lim ( ) ( )n
i i in
i
f x g x x
Which becomes ( ) ( )b
af x g x dx
Where a is the x-coordinate of the left intersection point
and b is the x-coordinate of the right intersection point
(topbottom)(little bit of width)
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The general process
1. Sketch your region.
2. If left and right bounds arent provided, find the
x-coordinate of the intersection points of thecurves.
3. Determine which curve is the upper curve andwhich curve is the lower curve.
4. Establish your integral
5. Evaluate the integral.
x-coordinate of right intersection point
x-coordinate of left intersection pointupper curve lower curve dx
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Example 12Find the area of the region bounded by 2 3y x and y x
1. Graph the region.
2. Find the x-coordinates of the
intersection points of the lines.
2
2
2
set the equations equal to each other and solve for x2 3
2 3 set equal to 0
2 3 0 factor
3 1 0 apply the zero product rule
3 1
y x
y x
x x
x x
x x
x or x
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3. Decide which equation is the
upper and which is the lower.
( ) 2 3f x x
2( )g x x
x = -1
x = 3
f(x)-g(x)
dx
4. Establish your rectangle fora representative area.
(height)(little bit of width)
(top-bottom)(little bit of width)
(f(x)g(x))dx
5. Establish and evaluate
your integral.
3 32 2
1 1
32 3
1
2 3 2 3
2
32 3
19 9 9 1 3
3
210 square units
3
x x dx x x dx
x x
x
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Taking TurnsSometimes the curves take turns being on the top and bottom.
Split the problem into 2 parts
and add the areas together.
Area of the yellow region
Area of the green region
Total area
(a, f(a))
(b, f(b))
(c, f(c))
f(x)
g(x)
( ) ( )b
af x g x dx
( ) ( )c
bg x f x dx
( ) ( ) ( ) ( )b c
a bf x g x dx g x f x dx
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Example 23 2Find the area of the region bounded by ( ) 4 ( ) 2f x x x and g x x x
1. Graph the region
2. Find the x-coordinates of the
intersection points
3 2
3 2
2
( ) ( )
4 2
6 0
6 0
3 2 0
0, 3, 2
f x g x
x x x x
x x x
x x x
x x x
x or x or x
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3( ) 4f x x x
2( ) 2g x x x
x = -3
x = 2
x = 0
dx
f(x)g(x)
dx
g(x)f(x)
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3. Establish your representative rectangles (see diagram)
4. Establish your integrals
0 2
3 0
0 23 2 2 3
3 0
0 23 2 3 2
3 0
( ) ( )
( ) ( ) ( ) ( )
4 2 2 4
6 6
right right
left left top bottom dx top bottom dx
f x g x dx g x f x dx
x x x x dx x x x x dx
x x x dx x x x dx
5. Evaluate your integrals.
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Going Sideways
Sometimes a curve is the upper and lower curve at the same
time. When this happens, we make horizontal rectangles instead
of vertical rectangles.
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A Sideways RectangleThe area is still height times
width.
This time well call the width
The height will still be the
upper curvethe lower curve,
but we will call the right most
curve the upper curve and the
left most curve the lower curve.
The curves must be solved for
x so that x = f(y).
The area becomes
y
( ) ( )f y g y y
y
x = f(y)
x = g(y)
f(y)g(y)(f(a), a)
(f(b), b)
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Still Sideways
1. Graph the region.
2. Solve each equation for x. This give you x = f(y).
3. Find the y-coordinate of the intersection points.
4. f(y) is the right most curve. g(y) is the left most curve.
5. Your integral is upper y
lower y
( ) ( )f y g y dy
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Example 32Find the area of the region bounded by 3 1x y and y x
1. Graph the region.
2. Solve the equations for x.
2
2
3 1
3 1
x y and y x
x y and x y
3. Find the y-coordinate
of the intersection points
2
2
3 1
0 2
0 2 1
2 1
y y
y y
y y
y or y
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y = 1
y = -2
f(y)
g(y)
dy
rightleft
f(x)g(x)
4. Establish your rectangle
5. Establish and evaluate
your integral
upper y
lower y
12
2
12
2
13 2
2
( ) ( )
3 1
2
23 2
1 1 82 2 4
3 2 3
9square units
2
f y g y dy
y y dy
y y dy
y yy