AP Calculus AB/BC · Unit 6 · Topic 6.6
Applying Properties of Definite Integrals
Use orientation, interval addition, linearity, and symmetry to combine known integrals.
1. Topic Focus
Interpret definite integrals as accumulated change, connect sums to integrals, apply both Fundamental Theorems, and select antiderivative techniques.
This topic: Use orientation, interval addition, linearity, and symmetry to combine known integrals.
2. Key Relationship
Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.
3. Visual Connection
4. Worked Example
Split an integral at a point where tabulated information changes.
Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.
5. Concept Development
A definite integral is a signed number
The value \(\int_a^b f(x)\,dx\) represents net signed accumulation from \(a\) to \(b\). It is positive when above-axis contributions dominate and negative when below-axis contributions dominate.
Equal bounds give zero
An interval with no width contributes no accumulation, regardless of the value of \(f(a)\).
Reversing bounds reverses orientation
The geometric region is unchanged, but traversing it in the opposite direction changes the sign.
Adjacent intervals add
This property also allows an unknown subinterval integral to be recovered by subtraction.
Constant multiples factor out
Multiplying every height by \(k\) multiplies the signed accumulation by the same factor.
Integrals are linear
Linearity applies when the component integrals use the same bounds and orientation.
Constant functions form rectangles
This is signed rectangle area and is often the missing term in a linear combination.
Geometry can evaluate exact values
When a graph consists of lines, triangles, rectangles, trapezoids, or circular arcs, calculate each geometric area exactly and attach a positive or negative sign according to its position relative to the axis.
Net area and total area differ
If geometric areas above and below the axis are \(A_+\) and \(A_-\), then
Properties of signed integrals do not automatically calculate total geometric area.
Symmetry can simplify an integral
On \([-a,a]\), an odd function has integral zero, while an even function satisfies
Verify the function's parity and the symmetry of the bounds before using these shortcuts.
Positive functions have nonnegative integrals
If \(f(x)\ge0\) on \([a,b]\), then \(\int_a^b f\ge0\). More generally, if \(f\le g\) throughout the interval, then \(\int_a^b f\le\int_a^b g\).
Bounds on a function bound its integral
If \(m\le f(x)\le M\) on \([a,b]\), then
This provides a reasonableness check even when the exact value is unknown.
Finite removable or jump discontinuities may be integrable
Continuity guarantees integrability, but it is not necessary. Changing finitely many point values does not change a definite integral, and a bounded function with finitely many jumps can still be integrated by splitting the interval.
The variable of integration is interchangeable
The expressions \(\int_a^b f(x)\,dx\) and \(\int_a^b f(t)\,dt\) are equal. The internal variable may be renamed consistently without changing the bounds or value.
A reliable property workflow
Align interval directions, join or split adjacent intervals, apply linearity, and only then insert known values. For graph problems, partition at axis crossings and shape boundaries before calculating geometry.
Common errors
Frequent errors include reversing bounds without changing sign, adding integrals over nonmatching intervals, treating below-axis area as positive, pulling a variable factor outside as though it were constant, and confusing signed integral with total area.
6. Detailed Worked Example and Error Check
Example 1: Combine known integrals. Suppose \(\int_0^2f=5\), \(\int_2^5f=-3\), \(\int_0^2g=1\), and \(\int_2^5g=4\). Then \(\int_0^5f=2\) and \(\int_0^5g=5\), so
Example 2: Recover a missing interval. If \(\int_{-2}^{3}f=7\) and \(\int_{-2}^{1}f=4\), then
Example 3: Geometry with both signs. A semicircle of radius 2 lies above the axis on \([-2,2]\), followed by a triangle of base 3 and height 4 below the axis on \([2,5]\). Therefore
The total geometric area is instead \(2\pi+6\).
Example 4: Symmetry. The function \(p(x)=x^3+2x\) is odd, so \(\int_{-4}^{4}p(x)\,dx=0\). The function \(q(x)=x^2+1\) is even, so \(\int_{-4}^{4}q(x)\,dx=2\int_0^4q(x)\,dx\).
Example 5: Constant contribution.
Example 6: Changing one point. Suppose \(h(x)=f(x)\) everywhere on \([0,6]\) except \(h(2)=100\). If \(f\) is integrable, then \(\int_0^6h=\int_0^6f\), because one point has zero interval width and contributes no area.
Example 7: Bound the answer. If \(2\le r(x)\le5\) on \([1,4]\), then
Any proposed value outside this range must be incorrect.
7. AP Reasoning Routine
Identify the accumulating quantity and units, preserve bounds, choose a valid integration technique, and check answers by differentiation.
- Identify the representation and requested quantity.
- State the rule or theorem and verify its conditions.
- Keep exact values until the final requested approximation.
- Interpret sign, units, interval, and context.
Apply integral properties without finding antiderivatives unless necessary.
(a) If \(\int_0^3f=4\) and \(\int_3^7f=-6\), find \(\int_7^0f\).
(b) If \(\int_1^5f=3\) and \(\int_1^5g=-2\), find \(\int_1^5(4f-3g)\,dx\).
(c) If \(\int_{-1}^{6}h=10\) and \(\int_2^6h=7\), find \(\int_{-1}^{2}h\).
(d) Simplify \(\int_4^4p(x)\,dx+\int_7^4p(x)\,dx\).
(e) A semicircle of radius 3 lies entirely below the axis on \([-3,3]\). Find its definite integral.
(f) If \(f\) is even and \(\int_0^5f=8\), find \(\int_{-5}^{5}f\).
(g) If \(g\) is odd, find \(\int_{-9}^{9}g(x)\,dx\).
(h) A graph has geometric area 4 above the axis and 7 below it. Find its signed integral and total area.
(i) Explain why changing the value of an integrable function at one input does not change its definite integral.
(j) If \(-2\le f(x)\le3\) on \([4,8]\), bound \(\int_4^8f(x)\,dx\).
Check the solution
(a) \(\int_0^7f=4-6=-2\), so \(\int_7^0f=2\).
(b) By linearity, the value is \(4(3)-3(-2)=18\).
(c) \(10=\int_{-1}^{2}h+7\), so the missing integral is \(3\).
(d) The first integral is zero and the second reverses orientation, giving \(-\int_4^7p(x)\,dx\).
(e) The geometric area is \(\frac12\pi(3)^2=9\pi/2\); because it lies below the axis, the integral is \(-9\pi/2\).
(f) Even symmetry gives \(2(8)=16\).
(g) Odd symmetry over a symmetric interval gives \(0\).
(h) The signed integral is \(4-7=-3\), while total area is \(4+7=11\).
(i) A single point has zero width in the Riemann-sum limit, so it contributes no accumulated area.
(j) Multiply the function bounds by the interval length 4: \(-8\le\int_4^8f(x)\,dx\le12\).