Integration

Suppose \(f\) is a bounded, nonnegative function on the interval \([a,b]\), and suppose it is possible to assign an area \(I\) underneath the graph. Given a partition \(P\) of \([a,b]\) into subintervals, erect in each subinterval two rectangles: one with height below the graph and one with height above the graph. Since we know the area of rectangles, we in this way get lower bounds \(L(f,P)\) and upper bounds \(U(f,P)\) on \(I\). This reasoning paves the way for how to define when "the area under the graph" exists, and what it in that case should be:
Definition 3: The definite integral
Suppose there is exactly one number \(I\) such that for every partition \(P\) of \([a,b]\) we have \[ L(f,P) \le I \le U(f,P).\] Then we say that the function \(f\) is integrable on \([a,b]\), and we call \(I\) the definite integral of \(f\) on \([a,b]\), denoted by the symbol \[I = \int_a^b f(x) dx.\] Remark: The above case adresses nonnegative functions, but may be extended to negative functions by allowing for negative areas. It is furthermore possible to show that every continuous (and every piecewise continuous) function is integrable.

The following properties of the Riemann integral belong to the most important ones, and should be learnt by heart.

Theorem 3
Let \(f\) and \(g\) be integrable on an interval containing the points \(a\), \(b\) and \(c\). Then the following holds:

a) An integral over an interval of zero length is zero:

\[ \int_a^a f(x)dx=0.\]

b) Reversing the limits of integration changes the sign:

\[\int_b^a f(x) dx=-\int_a^b f(x) dx.\]

c) Integration is linear: if \(A\) and \(B\) are constants, then

\[\int_a^b (Af(x)+Bg(x))dx=A\int_a^b f(x) dx+B\int_a^b g(x) dx.\]

d) The integral depends additively on the interval of integration:

\[\int_a^b f(x)dx +\int_b^c f(x) dx=\int_a^c f(x) dx.\]

e) The definite integral is order preserving, that is, if \(a<b\) and \(f(x)\leq g(x)\) for all \(x\in (a,b)\) then

\[ \int_a^b f(x) dx \leq \int_a^b g(x) dx.\]

f) The triangle inequality for integrals holds:

\[ \left| \int_a^b f(x) dx \right|\leq \int_a^b |f(x)| dx.\]

Solving integrals using the definition can be very arduous. The following result simplifies matters considerably, and is, as the name suggests, one of the most important result in calculus.

Theorem 5
Let \(f\) be a continuous function on an interval \(I\) containing the point \(a\). Define the function \(F\) on \(I\) by

\[F(x)=\int_a^x f(t) dt.\]

Then \(F\) is differentiable on \(I\) (with one-sided derivatives at the endpoints, if the interval is open), and \(F'(x)=f(x)\) on \(I\). \(F\) is therefore called an antiderivative of \(f\) on \(I\).

Furthermore, if \(G\) is any antiderivative of \(f\) on \(I\), then the following holds for any \(b\) in \(I\)

\[\int_a^b f(x) dx=G(b)-G(a).\]

The Fundamental Theorem of Calculus tells us that to integrate a function, it suffices to find an antiderivative of that function. Viewing the rules of differentiation as statements about antiderivatives, we acquire a range of integration techniques. Viewing the chain rule for differentiation in the same way, one is led to the method of substitution.

Theorem 6: Substitution in a definite integral
Suppose that \(g\) is a differentiable function on \([a,b]\). If the function \(f\) is continuous, then \[\int_a^b f(g(x)) g'(x) dx = \int_{g(a)}^{g(b)} f(u) du.\]
Remark: substitution also works on indefinite integrals, in which case one doesn't have to worry about changing the limits of integration.