Applying Properties of Definite Integrals
Use interval addition, reversal, linearity, symmetry, and bounded discontinuity properties before calculating.
College Board Topic 6.6 · CED p. 118 (PDF p. 123) · CED effective Fall 2020; project course outline 2026–27.
Use structure before doing arithmetic. Integrals are linear: ∫(cf+dg)=c∫f+d∫g. Adjacent intervals add: ∫ₐᶜ f + ∫꜀ᵇ f = ∫ₐᵇ f. Reversing limits changes the sign, and equal limits give zero. A semicircle can often be handled with geometry rather than a difficult antiderivative.
On a symmetric interval, an odd integrable function has integral zero; an even function has twice its integral over the positive half. Finitely many removable or jump discontinuities do not destroy integrability for otherwise piecewise continuous, bounded functions. Changing a value at a single point does not change the integral. An unbounded discontinuity needs separate improper-integral treatment in BC; it is not the same situation.
Let's work through it
- Given ∫₀² f=3 and ∫₂⁵ f=−1, obtain ∫₀⁵ f=2. Reason: Additivity joins adjacent intervals.
- Then ∫₅⁰ 2f=−4. Reason: Reversal gives −2 and the constant factor multiplies that result by 2.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
A bounded step function jumps once. Can its definite integral still exist?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
A bounded step function jumps once. Can its definite integral still exist?
Worked answer and reason
Yes.
Piecewise continuity with a finite jump allows the integral.
Retrieval 2
Changing f(1) from 2 to 200 changes ∫₀² f by how much?
Worked answer and reason
Zero.
A single point has zero width.
Catch the tempting mistake
- A discontinuous function is never integrable.
- A finite jump or removable discontinuity can still give an integral. Distinguish bounded jumps from an unbounded singularity.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 10 · No calculator · analytic
- Unit 6 problem 11 · No calculator · graphical
- Unit 6 problem 12 · No calculator · multiple
For the teacher
- Use a semicircle and a step graph, then an algebraic interval puzzle. Ask students to name the property used in every line.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
-
Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
-
Antiderivative
An antiderivative of a function is another function whose derivative equals the original function on an interval. Two antiderivatives on the same interval differ by a constant. For example, x² is an antiderivative of 2x, and x²+7 is another. An antiderivative is a function, whereas a definite integral is an accumulated value.
-
Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
-
Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.