Consider the interpretation of a definite integral in terms of area.

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Consider the interpretation of a definite integral in terms of area. (i) State the definition of the definite integral of a continuous function

f(x)

over the interval

[a,b]

using a sum. (ii) Using your answer in part (i), explain why the geometric area bounded by the graph of

y=f(x)

(where

f(x)\ge 0

[a,b]

) and the

x

-axis is given by

\int_a^b f(x)\,dx

. (iii) For the function

p(x)=-x

on the interval

[0,2]

, evaluate

\displaystyle\int_0^2 p(x)\,dx

and describe how this numerical value differs from the geometric area enclosed by the graph of

p(x)

and the

x

-axis on the same interval. Give reasons for your answer.