A continuous function is defined on the closed interval

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

A continuous function

h

is defined on the closed interval

[-2,4]

. It is known that

\begin{aligned} \text{(i)} & \quad \int_{-2}^{1} h(x)\,dx = 5,\\ \text{(ii)} & \quad \int_{1}^{4} h(x)\,dx = -3. \end{aligned}

The graph of

h

is entirely above the

x

-axis on

(-2,1)

and entirely below the

x

-axis on

(1,4)

. (1) Part 1: (a) Describe, without performing any additional calculations, the geometric meaning of the positive value in statement (i) and the negative value in statement (ii). (2) Part 2: (b) Calculate

\displaystyle \int_{-2}^{4} h(x)\,dx

and explain how its value relates to the net signed area enclosed by the graph of

h

and the

x

-axis on

[-2,4]

. State whether the total unsigned (geometric) area enclosed by the graph and the

x

-axis is greater than, equal to, or less than the absolute value of your answer, and justify your conclusion.

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