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Integration
Adrian Down
January 19, 2006
1 Riemann integral
1.1 Definition
Consider the graph of a function f between the points x = a and x = b. Subdivide the interval into n subintervals, all of equal width ∆x = b−na. Consider an arbitrary point zi within each subdivision. It can be proven using analysis that
lim n→∞
∑^ n
i=
f (zj )∆x
exists for f continuous in [a, b]. It is called the Riemann integral of f on [a, b]. It is traditionally notated
∫ (^) b
a
f (x)dx
Note. • dx is notational convenience meant to recall the differential change in x. The only information required to perform the integral is the func- tion f and the endpoints. Sometimes the dx is omitted.
- The x in the integral is a “dummy variable.” It can be replaced by any symbol, and the meaning of the integral is unchanged. It is good practice to have different labels for the limits of integration and the dummy variable.
1.2 Properties
1.2.1 Orientation
If the limits of integration are reversed, the integral changes sign.
∫ (^) a
b
f (x)dx = lim n→∞
∑^ n
i=
f (zn+1−i)
a − b n
The order of the summation doesn’t matter, so it can be reversed. The second term is −∆x.
∫ (^) a
b
f (x)dx = − lim n→∞
∑^ n
j=
f (zj )∆x
We have used the change of variables j = n + 1 − i. Recognize the definition of the derivative.
− lim n→∞
∑^ n
j=
f (zj )∆x = −
∫ (^) b
a
f (x)dx
1.2.2 Additivity
The integral can be subdivided and the separate integrals can be added.
∫ (^) b
a
f dx =
∫ (^) b
a
f dx +
∫ (^) c
b
f dx
2 Fundamental Theorem of Calculus
2.1 Theorem
Theorem (Fundamental Theorem of Calculus). Given f (x) continuous in [a, b], and F (x) is any antiderivative of f (x) (any function F such that F ′(x) = f (x) in (a, b)), then
∫ (^) b
a
f (x) = F (b) − F (a)
Applying the FTC to F
g(x)
as an anti-derivative,
F (g(b)) − F (g(a)) =
∫ (^) b
a
f (g(x))g′(x)dx
The endpoints and dummy variable can be relabeled, B = g(b) and A = g(a).
F (B) − F (A) =
∫ B
A
f (u)du
We have the substitution formula, ∫ (^) g(b)
g(a)
f (u)du =
∫ (^) b
a
f (g(x))g′(x)dx
Note. In the multidimensional case, the derivative is replaced by the Jaco- bian.
Example. ∫ π 2
0
cos x √ x
dx
Note. This is an improper integral, since the integrand is not defined at x = 0. However, the integral converges.
The Riemann sum converges slowly due to the √^1 x near 0. Use the substitution rule to obtain a function whose Riemann sums con- verges more rapidly. Choose u to simplify,
du =
dx √ x
⇒ u = 2
x
We also need x as a function of u,
x =
(u
2
u^2 4
The substitution formula then gives ∫ √ 2 π
0
cos
u^2 4
du
The graph is now a horizontal line near x = 0 and curves downward towards to intersect the x axis at x =
2 π. The Riemann sums for the new graph curve converge much more quickly, since there are no sharp contours in the graph.
2.2.2 Integration by parts
Integration by parts results from the backwards product of differentiation. From the chain rule,
(uv)′(x) = u(x)v′(x) + v(x)u′(x)
Performing the definite integral of both sides,
∫ (^) b
a
(uv)′dx =
∫ (^) b
a
uv′dx +
∫ (^) b
a
vu′dx
By the FTC,
∫ (^) b
a
(uv)′dx = [uv]ba
Rearranging terms,
∫ (^) b
a
uv′dx = [uv]ba −
∫ (^) b
a
vu′dx
Note. Integration by parts is the foundation of Taylor series. This connection is vital for numerical analysis.
3 Taylor series
Theorem. Let f (x) by analytic about x = a (i.e. f has derivatives of all orders in some interval about a). Then ∃R > 0 such that
f (x) =
∑^ ∞
n=
f (n)(a) n!
(x − a)n^ = lim N →∞
∑^ N
n=
f (n)(a) n!
(x − a)n
in |x − a| < R.
Note. There are two aspects to Taylor series: how to compute them and how to use them. Many elementary functions can be obtained using geometric series.
An alternate formulation of Taylor series in the spirit of numerical analysis is as follows:
