7.1 area between curves
TRANSCRIPT
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Area Between Curves
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The Big Picture
= -
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“Proof”
Find the limit of the area of rectangles given by a representative rectangle:
The area would be the limit of the sum of the areas of infinite representative rectangles with infinitely small intervals (n → ∞):
n
iii
nxxgxf
1
lim
This is the integral:
b
a
dxxgxf
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Steps, Hints, & Tricks
If not given, find the points of intersection by setting equations equal to each other. These are your interval values (a and b)
Graphs may intersect in more than 2 points – find all that apply
If a graph is a function of y, then change your perspective horizontally (a and b would be y values)
If the graphs switch relative position (top-bottom or left-right), then you must break the integral into separate integrals (see above graph)
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Example: Find the area between the curves
0 1
1
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Example: Find the area between the curves
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Area Between Curves
2
1
2
1
y
y
x
x
dyleftrightA
dxbottomtopA
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Example: Find the area between the curves
2
0 4 -1