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 a positive function over any interval is always positive.

Accepted Answer

False. If the function is zero almost everywhere (e.g., zero except at a point), the integral is zero, not positive.

Question 3

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 4

True or false: There exist continuous functions that are not Riemann integrable.

Accepted Answer

False. Every continuous function on a closed interval is Riemann integrable. The Dirichlet function is discontinuous and non-integrable.

Question 5

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 6

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 7

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.

Question 8

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

True. By Cauchy's integral theorem, analytic functions have zero contour integrals over closed curves, a core result in complex analysis.

INTEGRAL Trivia Questions