Question

The following figure shows the graph of a continuous function $y=f(x)$ on the interval $[1,3]$. The points $A,B,C$ have coordinates $(1,1),(3,2),(2,3)$ respectively, and the lines $l_1$ and $l_2$ are parallel, with $l_1$ being tangent to the curve at $C$. If the area under the graph of $y=f(x)$ from $x=1$ to $x=3$ is $4$ sq units, then the area of the shaded region is

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

Answer: (A)

Given,
$A(1,1),B(3,2),C(2,3)$

Slope of line $l_2=\frac{2-1}{3-1}=\frac{1}{2}$
Line $l_1$ is parallel to $l_2$
$\therefore$ Equation of line $l_1$ is
$\Rightarrow y-3=\frac{1}{2}(x-2)$
$\Rightarrow y=\frac{x}{2}+2$
Area of shaded region
$=\underset{1}{\overset{3}{\int }}$ (line $l_1$ -curve $f(x))dx$
$=\underset{1}{\overset{3}{\int }}\left(\frac{x}{2}+2\right)dx-4$
$[\because$ area of curve = 4]
$={\left[\frac{x^2}{4}+2x\right]}_1^3-4$
$=\left(\frac{9}{4}+6\right)-\left(\frac{1}{4}+2\right)-4$
$=\frac{9}{4}+6-\frac{1}{4}-2-4=2$