Answer

Verified

418.8k+ views

**Hint:**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} $ .

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE