Johann Carl Friedrich Gauss 1777 – 1855 Johann Carl Friedrich Gauss 1777 – 1855 Gauss worked in...
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Johann Carl Friedrich Gauss 1777 – 1855 Gauss worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. His work has had an immense influence in many areas.
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Transcript of Johann Carl Friedrich Gauss 1777 – 1855 Johann Carl Friedrich Gauss 1777 – 1855 Gauss worked in...
Johann Carl Friedrich Gauss1777 – 1855
Gauss worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. His work has had an immense influence in many areas.
2( ) 2f x x
( )g x x
2( ) 2f x x
( )g x x
Consider a very thin vertical strip or rectangle.
The length of the strip is:
( ) ( )f x g x or 22 x x
Since the width of the strip is a very small change in x, we could call it dx.
x
2( ) 2f x x
( )g x x
( )f x
( )g x
( ) ( )f x g xdx
Since the strip is a long thin rectangle, the area of the strip is:
2length width 2 x x dx
If we add all the strips, we get:2 2
1(2 ) x x dx
x
2( ) 2f x x
( )g x x
2 2
1(2 ) x x dx
23 2
1
1 123 2
x x x
8 1 14 2 23 3 2
8 1 16 23 3 2
36 16 12 2 36
276
29 units2
x
y x
2y x
y x
2y x
If we try vertical strips, we have to integrate in two parts:
dx
dx
2 4
0 2 2x dx x x dx
We can find the same area using a horizontal strip.
dySince the width of the strip is dy, we find the length of the strip by solving for x in terms of y.y x
2y x
2y x
2y x
x x
y
y x
2y x
We can find the same area using a horizontal strip.
dySince the width of the strip is dy, we find the length of the strip by solving for x in terms of y.y x
2y x
2y x
2y x
2 2
02 y y dy
length of strip
width of strip
22 3
0
1 122 3y y y
82 43
210 units3
y
General Strategy for Area Between Curves:
1 Sketch the curves.
Decide on vertical or horizontal strips connecting the boundary curves. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.)
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.
Examples:21) Find the area of the region enclosed by the curves 5 0, ,
1, and 2.x y x y
y y
2) Find the area of the region enclosed by the curves sin and sin 2 over the interval 0, .
y xy x
3) Find the area of the region enclosed by the curves below.
