Area Between Curves By Shifting Of Origin

Area Between Curves By Shifting Of Origin

While calculating area of a region using definite integral. Sometimes it is more convenient to evaluate the required area by shifting the given curves with respect to the origin. It may be seen that the original functions are given in their general form and their manipulation, in order to find the area, is a little bit lengthy.

Although if we shift the curve with respect to the origin. Then we get the equation of these curves in their standard form. So it could be much easier to manipulate the shifted curves to find the area. Now in this tutorial, we will be focusing on such a method of finding the area of the region with the help of Definite Integral by shifting the origin.

Note: If we shift the curve anywhere in x-y plane. Then the area of the region remains the same across the plane. So this concept is very important to solve the given question in an easier way.

Now we will be taking the help of some solved examples in order to understand the concept in a better way.

Example: Find the area of the region bounded by the curves (x-1)2 = 3(y-1) and (y-1)2 = 3(x-1).
Solution: Let us take (x-1) = X and (y-1) = Y. Now the equations of given curves will be. X2 = 3Y and Y2 = 3X. These two curves intersect each other at (0, 0) and (3, 3). Now the area enclosed by these two curves will be given by

Area=\left| \int ^{3}_{0}\left( \dfrac{X^{2}}{3}-\sqrt{3X}\right) dX\right| =3

Example: Find the area of the region bounded by the curves (y-2) = e(x-1), y-2 = e(1-x) and the straight line x = 3.
Solution: Let us take y-2 = Y and x-1 = X. Now we have to calculate area of the region bounded by the curves Y = eX, Y = e-X and the straight line X = 2. Thus the required area is given by \begin{aligned}Area=\int ^{2}_{0}\left( e^{X}-e^{-X}\right) dX=\left( e^{X}+e^{-X}\right) _{0}^{2}\\ \Rightarrow Area=e^{2}+\dfrac{1}{e^{2}}-2\end{aligned}

Now attempt at least five questions based on this topic. So this is it from this tutorial. Hoping you people will attempt a few problems based on the topic discussed in this tutorial. In the next tutorial, we will discuss the topic “Area Bounded By Closed Curves”.

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