Function is continuous on

MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

Function

p

is continuous on

[0,5]

, positive on

(0,2)

, and negative on

(2,5)

. Which of the following best describes the value of

\displaystyle \int_{0}^{5} p(x)\,dx

? (a) It equals the sum of the absolute areas of the two regions, so it must be positive. (b) It is negative because the interval where

p

is below the axis is longer, so the negative area must outweigh the positive. (c) It equals the area above the

x

–axis on

(0,2)

minus the area below the

x

–axis on

(2,5)

. (d) It is zero because the positive and negative areas cancel whenever the function changes sign.