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4.1 Area problem
• We know how to compute areas of rectilinear objects, such as rectangles, triangles, polygons
• How do we define and compute areas of more complicated regions (e.g. area enclosed by a circle)?
• Idea: approximate such regions by rectilinear regions (for example, by polygons)
Area under the curve y=f(x) between a and b
x
y
a
y = f(x)
bx
Assume f(x) ≥0 on [a,b] and consider regionR = { (x,y) | a ≤x ≤ b, 0 ≤ y ≤ f(x) }
(x,y)
f(x)
What is the area of R?
=xnx1 x2xi-1 xi
Approximation by rectangles
x
y
a
y = f(x)
bx0=
• Divide [a,b] into n intervals of equal length
• Use right endpoints to built rectangles (columns)
Total area of all columns is
x
y
a
y = f(x)
bx0= =xnx1 x2xi-1 xi
f(xi)
∆x
f(xi)∆x
f(x2)∆xf(x1)∆x
f(xn)∆x
xxfxxfxxfxxf ni )(...)(...)()( 21
xxfRn
iin
1
)(
Theorem. If f is continuous on [a,b] then the following limit exists:
x
y
a
y = f(x)
b
xxfAn
ii
n
1
)(lim
How to find xi
x
a bx0= =xnx1 x2xi-1 xi
∆x ∆x ∆x
nabaxaxxax
nabaxaxax
axoax
/)(22
/)(1
2
1
0
nabiaxiaxi /)(
nabx /)(
=xnx1 x2 xi-1xi
x
y
a
y = f(x)
bx0=
f(xi-1)
∆x
Total area of all columns is
xxfxxfxxfxxf ni )(...)(...)()( 1110
xxfLn
iin
11)(
Nevertheless…
• Theorem. If f(x) is continuous on [a,b], then both limits and
exist and
nnR
lim n
nL
lim
nn
nn
LR
limlim
x*nx*ix*2
Using sample points
=xnx1 x2xi-1 xi
x
y
a
y = f(x)
bx0=
Choose a sample point - an arbitrary point xi* in [xi-1, xi]for each i
x*1
Total area of all columns is
xxfxxfxxfxxf ni )(...)(...)()( ***2
*1
xxfn
ii
1
*)(
x*i=xnx1 x2
xi-1 xi
x
y
a
y = f(x)
bx0= x*1 x*2 x*n
Theorem
• If f(x) is continuous on [a,b], then the limit
exists and does not depend on the choice of sample points
xxfn
ii
n
1
*)(lim
4.2 Definite Integral
• Now we consider functions that may change sign on [a,b]
• In this case, we need to take into account sign of f(x)
• Idea: use “signed area”
Net area of all columns is
xxfxxfxxfxxf ni )(...)(...)()( ***2
*1
xxfn
ii
1
*)(
x*n
x*i
x*2=xnx1 x2
xi-1 xi x
y
a
y = f(x)
bx0= x*1
Riemann Sum that correspond ton and given choice of sample points
xxfn
ii
1
*)(
x*n
x*i
x*2=xnx1 x2
xi-1 xi x
y
a
y = f(x)
bx0= x*1
Definite Integral of function f from a to b is defined as the limit of Riemann sums
xxfdxxfn
ii
b
an
1
*)(lim)(
x*n
x*i
x*2=xnx1 x2
xi-1 xi x
y
a
y = f(x)
bx0= x*1
Theorem
• If f(x) is continuous on [a,b], then the definite integral of function f from a to b
exists and does not depend on the choice of sample points
xxfdxxfn
ii
b
an
1
*)(lim)(