A Strategy for Mixed Integral Problems
Many definite-integral problems are not solved by applying one property in isolation. A useful solution may require splitting an interval, using symmetry on part of it, applying linearity, and then estimating what remains. The key is to keep each step tied to its assumptions and to track the orientation of every interval.
Throughout this tutorial, the integrals are Riemann integrals of integrable functions on the intervals in question. Linearity and additivity apply to such integrals. Symmetry statements require the relevant evenness or oddness condition on a symmetric interval. Comparison requires a pointwise inequality throughout the interval.
Properties to Combine
For integrable functions \(f\) and \(g\), constants \(a\) and \(b\), and \(u\leq v\leq w\), linearity and additivity give
Orientation is recorded by \(\int_u^v f(x)\,dx=-\int_v^u f(x)\,dx\), and \(\int_u^u f(x)\,dx=0\). If \(f\) is even, meaning \(f(-x)=f(x)\), then \(\int_{-a}^{a}f(x)\,dx=2\int_0^a f(x)\,dx\). If \(f\) is odd, meaning \(f(-x)=-f(x)\), then \(\int_{-a}^{a}f(x)\,dx=0\). These symmetry rules apply only on intervals symmetric about zero.
Comparison and enclosure rules convert pointwise information into bounds. If \(f(x)\leq g(x)\) throughout \([a,b]\), where \(a\leq b\), then \(\int_a^b f(x)\,dx\leq\int_a^b g(x)\,dx\). In particular, if \(m\leq f(x)\leq M\) throughout the interval, then \(m(b-a)\leq\int_a^b f(x)\,dx\leq M(b-a)\).
Worked Example: Linearity, Additivity, Symmetry, and Orientation
Worked Example: Linearity, Additivity, Symmetry, and Orientation
Suppose \(f\) is even on \([-2,2]\), \(g\) is odd on \([-2,2]\), and \(\int_0^1 f(x)\,dx=1\) and \(\int_1^2 f(x)\,dx=2\). Evaluate \(\int_2^{-2}(2f(x)-3g(x))\,dx\).
First use additivity to combine the two known pieces of the integral of \(f\):
Since \(f\) is even, its integral over the symmetric interval is twice its integral over the positive half:
Since \(g\) is odd, its integral over the same symmetric interval is zero. Linearity now gives the integral in the forward direction:
The requested integral has its endpoints reversed, so change its sign:
Check: The forward integral is \(12\), and reversing its endpoints gives \(-12\), as required. The odd part contributes zero only because the interval is symmetric.
Worked Example: Comparison, Additivity, and a Reversed Bound
Worked Example: Comparison, Additivity, and a Reversed Bound
Let \(h\) be integrable, with \(0\leq h(x)\leq 2\) on \([0,1]\) and \(1\leq h(x)\leq 3\) on \([1,3]\). Find bounds for \(\int_3^0 h(x)\,dx\).
Estimate the two forward pieces separately. On \([0,1]\), the interval has length \(1\), so comparison with the constant functions \(0\) and \(2\) gives
On \([1,3]\), the interval has length \(2\). The lower and upper constant bounds give
By additivity, add the lower bounds and the upper bounds:
The requested direction is reversed. Negating the entire inequality reverses the inequality signs:
Check: The forward integral lies between \(2\) and \(8\), so its reversed version must lie between \(-8\) and \(-2\). Both bounds are negative, consistent with reversing the direction of integration for a nonnegative function.
Why Comparison Gives an Integral Bound
The comparison rule follows from the Darboux definition of the integral. Here is a short justification that also explains why the pointwise condition matters.
Define \(q(x)=g(x)-f(x)\). The hypotheses imply \(q(x)\geq0\) throughout the interval, and \(q\) is integrable because it is a difference of integrable functions. For any partition \(P\) of \([a,b]\), the infimum of \(q\) on each subinterval is nonnegative. Thus the lower Darboux sum \(L(q,P)\), formed by multiplying each such infimum by the subinterval's nonnegative width and adding, satisfies \(L(q,P)\geq0\). The integral is at least every lower Darboux sum, so \(\int_a^b q(x)\,dx\geq0\). By linearity,
Rearranging proves the comparison theorem. The proof depends on \(f(x)\leq g(x)\) at every point of the interval; a few sampled values do not establish the needed condition.
Worked Example: A Darboux Estimate Checked Geometrically
Worked Example: A Darboux Estimate Checked Geometrically
Estimate \(\int_0^2 x\,dx\) using the equal-width partition \(P=\{0,\tfrac12,1,\tfrac32,2\}\), then check the estimate by geometry.
The function \(f(x)=x\) is increasing on \([0,2]\). Therefore, on each subinterval the minimum occurs at the left endpoint and the maximum at the right endpoint. Each subinterval has width \(\tfrac12\). The lower Darboux sum is
The upper Darboux sum is
The Darboux bounds place the integral between these sums:
The graph \(y=x\) on \([0,2]\) encloses a right triangle with base \(2\) and height \(2\). Its area is \(\tfrac12(2)(2)=2\), so the geometric value is within the estimated bounds.
Check: \(\tfrac32\leq2\leq\tfrac52\). The lower rectangles lie beneath the graph and the upper rectangles cover it, consistent with the computed sums.
Worked Example: Symmetry and a Bounded Remainder
Worked Example: Symmetry and a Bounded Remainder
Suppose \(q\) is even and integrable on \([-3,3]\), \(\int_0^2q(x)\,dx=4\), and \(1\leq q(x)\leq2\) on \([2,3]\). Bound \(\int_{-3}^3q(x)\,dx\).
First bound the integral over the remaining unit interval. Comparison gives
By additivity and the given value on \([0,2]\),
Evenness allows the integral over \([-3,3]\) to be expressed as twice the integral over \([0,3]\). Multiplying the bounds by the positive number \(2\) preserves their direction:
Check: The positive-half integral is between \(5\) and \(6\); doubling gives the stated interval \([10,12]\). No information about \(q\) on \([2,3]\) was replaced by an unsupported exact value.
A Common Sign Mistake
A frequent error is to reverse the endpoints but leave an inequality unchanged. For example, from \(2\leq\int_0^3 h(x)\,dx\leq8\), one might incorrectly write \(2\leq\int_3^0 h(x)\,dx\leq8\). This contradicts \(\int_3^0h(x)\,dx=-\int_0^3h(x)\,dx\). Negating all three parts of the original inequality gives \(-2\geq\int_3^0h(x)\,dx\geq-8\), which, when written in increasing order, is \(-8\leq\int_3^0h(x)\,dx\leq-2\).
Before accepting a mixed estimate, check the interval direction, the sign of the integrand or its bounds, and whether the property used actually applies on the whole interval. In particular, symmetry cannot be used on a nonsymmetric interval without first separating a symmetric part, and a pointwise comparison cannot be inferred from a few plotted or sampled values.
Choosing and Checking a Method
Look for known subinterval integrals, even or odd symmetry, pointwise bounds, or upper and lower Darboux sums.
Split intervals at known points, separate sums using linearity, or isolate a symmetric interval. Record any reversal of endpoints.
Check that inequalities hold throughout each interval and account for its width when using constant or Darboux bounds.
Check the sign, interval lengths, arithmetic, and whether the final value or estimate is consistent with the geometric or symmetry information.
These checks are especially useful when several valid properties are combined: each individual step may be familiar, while a single missed sign or interval endpoint can invalidate the final conclusion.
Check Your Understanding
For each question, state which property or estimate is needed and keep the interval direction explicit.
- If \(f\) is even on \([-a,a]\), how can \(\int_{-a}^a f(x)\,dx\) be written using an integral over \([0,a]\)?
- If \(0\leq r(x)\leq4\) on \([1,3]\), what bounds follow for \(\int_3^1r(x)\,dx\)?
- Why does \(f(x)\leq g(x)\) throughout \([a,b]\) imply an inequality between their integrals?
- For an increasing function and a fixed partition, where on each subinterval are the minimum and maximum attained?
- What must change in an inequality when all three parts are multiplied by \(-1\)?
