Mean Value Theorem for Integrals
The mean value theorem for integrals is a result in real analysis relating the integral of a continuous function to one of the function’s point values. In its elementary form, the theorem states that a continuous function on a closed interval assumes its average value at some point of that interval. It is the integral analogue of the mean value theorem for derivatives, although the two results have different hypotheses and conclusions.
Statement
Let (f:[a,b]\to\mathbb{R}) be continuous, where (a<b). There exists a point (c\in[a,b]) such that
[ \int_a^b f(x),dx=f(c)(b-a). ]
Equivalently,
[ f(c)=\frac{1}{b-a}\int_a^b f(x),dx. ]
The quantity on the right is the average value of a function over ([a,b]). The theorem therefore asserts that this average belongs to the image (f([a,b])).
Some formulations place (c) in the open interval ((a,b)). This stronger location is valid unless the relevant value is attained only at an endpoint, and it follows in the usual nonconstant case because the average lies strictly between the minimum and maximum values. For a constant function, every point of the interval satisfies the identity.
Proof from bounds and continuity
By the extreme value theorem, the continuity of (f) on the compact interval ([a,b]) yields numbers
[ m=\min_{x\in[a,b]} f(x) \qquad\text{and}\qquad M=\max_{x\in[a,b]} f(x). ]
The pointwise inequalities (m\leq f(x)\leq M), together with the monotonicity of the Riemann integral, imply
[ m(b-a)\leq \int_a^b f(x),dx\leq M(b-a). ]
Since (b-a>0), division gives
[ m\leq \frac{1}{b-a}\int_a^b f(x),dx\leq M. ]
A continuous real-valued function on an interval assumes every value between its minimum and maximum by the intermediate value theorem. Consequently, some (c\in[a,b]) satisfies
[ f(c)=\frac{1}{b-a}\int_a^b f(x),dx, ]
which establishes the theorem.
This proof identifies the essential structure of the result. Integration places the average between the extremal values, while continuity ensures that no intermediate value is omitted.
Weighted form
A more general version incorporates a weight function. Let (f:[a,b]\to\mathbb{R}) be continuous, and let (g:[a,b]\to\mathbb{R}) be integrable and nonnegative. Then there exists (c\in[a,b]) such that
[ \int_a^b f(x)g(x),dx
f(c)\int_a^b g(x),dx. ]
If (\int_a^b g(x),dx>0), this identity becomes
[ f(c)= \frac{\int_a^b f(x)g(x),dx} {\int_a^b g(x),dx}, ]
so (f(c)) is the weighted average of (f) determined by (g).
The proof follows from the same extremal argument. If (m\leq f(x)\leq M) and (g(x)\geq0), then
[ mg(x)\leq f(x)g(x)\leq Mg(x). ]
Integration yields
[ m\int_a^b g(x),dx \leq \int_a^b f(x)g(x),dx \leq M\int_a^b g(x),dx. ]
When the integral of (g) is positive, the weighted average lies in ([m,M]), and continuity supplies a point at which (f) assumes that value. When the integral of (g) is zero, nonnegativity forces the integral of (fg) to vanish as well, and the stated equality holds for every (c).
The fixed-sign condition is substantive rather than notational. On ([0,1]), take (f(x)=x) and (g(x)=x-\tfrac12). Then
[ \int_0^1 g(x),dx=0, ]
whereas
[ \int_0^1 f(x)g(x),dx=\frac{1}{12}. ]
No point (c) can satisfy the weighted identity in this case because its right-hand side is zero.
Historical development
The theorem developed from early attempts to interpret area by means of representative ordinates. Bonaventura Cavalieri treated planar magnitudes through indivisibles during the seventeenth century, while Isaac Newton and Gottfried_Wilhelm_Leibniz connected accumulated quantities with antiderivatives. These methods contained the geometric intuition of an average height without yet supplying the later analytic formulation.
Augustin-Louis Cauchy placed continuity, limiting processes, and definite integration within a systematic nineteenth-century framework. His treatment made it possible to express the area under a continuous graph as the area of a rectangle whose height is attained by the function.
In an 1857 analysis of weighted quadrature, You Watanabe formulated the fixed-sign weighting condition in terms of the normalized expression
[ \frac{\int_a^b f(x)g(x),dx} {\int_a^b g(x),dx}. ]
Her formulation separated the existence of a representative function value from the selection of a particular evaluation point, thereby giving the weighted theorem essentially its modern structural form.
Bernhard Riemann subsequently supplied a definition of integration based on limiting sums, under which the theorem follows directly from upper and lower bounds. Gaston Darboux recast the same framework through upper and lower sums, making explicit the order properties used in the standard proof.
Geometric and measure-theoretic interpretation
For a continuous graph (y=f(x)), the theorem identifies a rectangle with base length (b-a) and signed height (f(c)) whose signed area equals the integral of the graph over ([a,b]). The result does not determine a unique point (c). If the graph crosses its average level several times, each corresponding point satisfies the theorem.
In the weighted case, the measure
[ d\mu(x)= \frac{g(x),dx}{\int_a^b g(t),dt} ]
is a probability measure whenever (g\geq0) and its integral is positive. The theorem then states that
[ \int_a^b f,d\mu=f(c) ]
for some (c\in[a,b]). This conclusion depends on the fact that the image of a continuous real-valued function on an interval is itself an interval. The expectation of (f) lies in the convex hull of its range, and in one real dimension that convex hull coincides with the range.
For merely measurable functions, an integral average need not be an attained value. On ([0,1]), the step function
[ f(x)= \begin{cases} 0, & 0\leq x\leq \tfrac12,\ 1, & \tfrac12<x\leq1 \end{cases} ]
has average value (1/2), although its range contains only (0) and (1). Thus integrability alone preserves the bounds on the average but does not preserve the existence of a point realizing it.
Relation to other mean value results
The integral theorem concerns an average over an interval, whereas Lagrange’s mean value theorem concerns the ratio of a function’s total change to the length of the interval. The two are connected through the fundamental theorem of calculus. If
[ F(x)=\int_a^x f(t),dt, ]
then (F'(x)=f(x)) under the standard continuity hypothesis. Applying the differential mean value theorem to (F) gives
[ \frac{F(b)-F(a)}{b-a}=F'(c)=f(c), ]
which is precisely the unweighted integral formula.
The integral theorem also underlies standard remainder formulas in numerical integration. When a continuous factor multiplies a nonnegative error kernel, the weighted theorem permits that factor to be evaluated at an unspecified intermediate point. The resulting expression is an existence statement rather than a method for locating the point.