Question

Area bounded by the curve $xy=c$ and the x-axis between $x=1$ and $x=4$ , is:
A. $c\log 3$ sq. units
B. $2\log c$ sq. units
C. $2c\log 2$ sq. units
D. $2c\log 5$ sq. units

Answer 1

The area under the function $y=f(x)$ from $x=a$ to $x=b$ and the x-axis is given by the definite integral $\left| \int_{a}^{b}{f}(x)\ dx \right|$ , for curves which are entirely on the same side of the x-axis in the given range.
If the curves are on both the sides of the x-axis, then we calculate the areas of both the sides separately and add them.
Definite integral: If $\int{f}(x)dx=g(x)+C$ , then $\int_{a}^{b}{f}(x)\ dx=[g(x)]_{a}^{b}=g(b)-g(a)$ .

Complete step-by-step answer:
The given equation of the curve is $xy=c$ which can also be written as $y=f(x)=\dfrac{c}{x}$ .
Using definite integrals, the area under the curve from $x=1$ to $x=4$ and the x-axis, will be given as:
$A=\left| \int_{1}^{4}{\dfrac{c}{x}}\ dx \right|$
Using $\int{\dfrac{1}{x}dx}=\log x+C$ , we get:
⇒ $A=c\left[ \log x \right]_{1}^{4}$
⇒ $A=c(\log 4-\log 1)$
Using $\log 1=0$ and $\log 4=\log {{2}^{2}}=2\log 2$ , we get:
⇒ $A=2c\log 2$ sq. units
The correct answer is C. $2c\log 2$ sq. units.

Note: The graph of $xy=c$ is a rectangular hyperbola.
In order to calculate the area of a curve from $y=a$ to $y=b$ and the y-axis, we will make use of $\left| \int_{a}^{b}{f}(y)\ dy \right|$ .
The length of a curve $y=f(x)$ from $x=a$ to $x=b$ is given by $L=\int_{a}^{b}{\sqrt{1+{{\left( \dfrac{dy}{dx} \right)}^{2}}}dx}$ .