You want to show that |∫[0,1] sin(x²) dx| ≤ 1. Which property of the Riemann integral justifies the key step?
ALinearity: ∫(af + bg) = a∫f + b∫g
BAdditivity: ∫[0,1] = ∫[0,c] + ∫[c,1] for any c ∈ (0,1)
CTriangle inequality: |∫f| ≤ ∫|f|
DMonotonicity: f ≤ g implies ∫f ≤ ∫g
The triangle inequality for integrals, |∫[a,b] f| ≤ ∫[a,b] |f|, is precisely the tool for bounding the absolute value of an integral. Here |∫[0,1] sin(x²) dx| ≤ ∫[0,1] |sin(x²)| dx ≤ ∫[0,1] 1 dx = 1. Monotonicity alone (option D) wouldn't let you move the absolute value inside the integral — it compares two integrals of different functions, not an absolute value of an integral to an integral of an absolute value.
Question 2 Multiple Choice
A function f has a single jump discontinuity at x = c ∈ (a, b) but is otherwise continuous and bounded. You want to integrate f over [a, b]. Which property is most directly useful?
ALinearity, because f can be written as a sum of simpler functions
BMonotonicity, because f is bounded above by a continuous function
CAdditivity over subintervals: ∫[a,b] f = ∫[a,c] f + ∫[c,b] f
DThe triangle inequality, to handle the absolute value at the discontinuity
Additivity over subintervals lets you split [a,b] at the discontinuity, dealing with [a,c] and [c,b] separately — on each subinterval f is continuous and therefore Riemann integrable. This is the standard strategy for handling isolated discontinuities: isolate them at endpoints of subintervals, where integrability is unaffected. Linearity (option A) applies to sums of functions, not to splitting domains.
Question 3 True / False
If f and g are Riemann integrable on [a, b] and f(x) ≤ g(x) for all x ∈ [a, b], then ∫[a,b] f ≤ ∫[a,b] g.
TTrue
FFalse
Answer: True
This is the monotonicity (order) property of the Riemann integral. It follows directly from the Darboux sum definition: if f(x) ≤ g(x) everywhere, then for any partition, every lower Darboux sum of f is ≤ the corresponding lower sum of g, and similarly for upper sums. Taking the limit gives ∫f ≤ ∫g. This property is used constantly to bound integrals by replacing a complicated integrand with a simpler upper bound.
Question 4 True / False
For any Riemann integrable function f on [a, b], the equality |∫[a,b] f| = ∫[a,b] |f| holds.
TTrue
FFalse
Answer: False
This confuses the triangle inequality with an equality. The correct statement is |∫[a,b] f| ≤ ∫[a,b] |f|, which is an inequality, not generally an equality. Equality holds only when f does not change sign on [a, b] — for instance, if f ≥ 0 everywhere, then |∫f| = ∫f = ∫|f|. But if f takes both positive and negative values, the integral ∫f involves cancellation, making it strictly smaller in absolute value than ∫|f|. For example, ∫[0, 2π] sin(x) dx = 0, but ∫[0, 2π] |sin(x)| dx = 4.
Question 5 Short Answer
Why does the linearity property ∫(af + bg) = a∫f + b∫g hold for Riemann integrals, and why is it useful?
Think about your answer, then reveal below.
Model answer: Linearity holds because the Riemann integral is a limit of Darboux sums, and sums are linear: the sum of af(xᵢ)Δxᵢ + bg(xᵢ)Δxᵢ equals a·(sum of f terms) + b·(sum of g terms). This additivity over summands carries through to the limit. It is useful because it lets you break a complicated integrand into simpler pieces — integrating each separately and combining results — without returning to the definition each time.
The key is that the integral inherits linearity from summation, since it is defined as a limit of sums. This is why polynomial integrals can be done term-by-term, why scaling a function scales its integral, and why subtraction of integrals corresponds to the integral of a difference. Linearity is the algebraic backbone that makes the Fundamental Theorem of Calculus and most integration techniques possible.