Question 1

True or false: The word 'integral' shares its Latin root with the word 'integer', meaning 'whole' or 'untouched'.

Accepted Answer

True. Both come from Latin 'integer' (whole), reflecting how integration sums parts to form a whole.

Question 2

True or false: The integral of x^(-1) from 1 to infinity equals 1.

Accepted Answer

False. The integral of 1/x diverges to infinity (harmonic series), as the antiderivative is ln(x), which grows without bound.

Question 3

True or false: The integral of a positive function over an interval with positive length is always positive.

Accepted Answer

True. A positive function has strictly positive values. Over an interval of positive length, its Riemann integral accumulates positive area, yielding a strictly positive result.

Question 4

True or false: The integral of e^(-x^2) from negative infinity to infinity equals the square root of pi.

Accepted Answer

True. This Gaussian integral is a classic result, often proven via polar coordinates or the gamma function.

Question 5

True or false: Every continuous function on a closed interval is Riemann integrable.

Accepted Answer

True. Continuous functions on a closed bounded interval are bounded and uniformly continuous, so their Riemann sums converge. This is a fundamental theorem in calculus.

Question 6

True or false: The integral of a function over a closed curve in the plane is always zero if the function is analytic.

Accepted Answer

False. False: Cauchy's integral theorem requires the function to be analytic in a simply connected domain. For example, f(z)=1/z is analytic on ℂ\{0} but its integral around the unit circle is 2πi, not zero.

Question 7

True or false: The fundamental theorem of calculus only applies to functions with a closed-form antiderivative.

Accepted Answer

False. The theorem applies to any continuous function, regardless of whether its antiderivative can be expressed elementarily.

Question 8

True or false: The Lebesgue integral can integrate functions that the Riemann integral cannot.

Accepted Answer

True. Lebesgue integration handles highly irregular functions (e.g., Dirichlet function) by measuring sets, not just intervals.

INTEGRAL Trivia Questions