On the interval

MATHEMATICS · Single correct · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

On the interval [0,1][0,1], which of the following statements is true about the lengths of the curves y=x2y=x^{2} and y=ex−1y=e^{x}-1? (a) The curve y=x2y=x^{2} is longer because 2x>ex2x>e^{x} for every xx in [0,1][0,1]. (b) The curve y=x2y=x^{2} is longer because ∫012x dx=1\displaystyle\int_{0}^{1}2x\,dx=1 is greater than ∫01ex dx≈0.718\displaystyle\int_{0}^{1}e^{x}\,dx\approx0.718. (c) The curve y=ex−1y=e^{x}-1 is longer because ex>2xe^{x}>2x for all xx in [0,1][0,1]. (d) The curve y=x2y=x^{2} is longer because x2>ex−1x^{2}>e^{x}-1 for every xx in [0,1][0,1].

Related theory

Browse the problem library