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
$=\int\limits_{1}^{3}$ (line $l_{1}$ -curve $f (x))dx $
$=\int\limits_{1}^{3}\left(\frac{x}{2}+2\right)dx-4$
$[\because$ area of curve = 4]
$=\left[\frac{x^{2}}{4}+2x\right]_{1}^{3}-4$
$=\left(\frac{9}{4}+6\right)-\left(\frac{1}{4}+2\right)-4$
$= \frac{9}{4}+6-\frac{1}{4}-2-4=2$