#### Area Under Curve Tutorial

## Area Bounded By Closed Curve

In this tutorial we will learn how to find the area of a region bounded by a closed curve. To do so consider the figure (1.1) given in the thumbnail.

Let PM = y_{1} and P’M = y_{2}. The area of the verical strip (given in the figure in thumbnail) is (y_{1} – y_{2}) dx. And the area of the region bounded by the closed curve will be Area=\int ^{b}_{a}\left( y_{1}-y_{2}\right) dx. Here OB = a and OB’ = b.

**Example**: Find the area bounded by closed curve y = (4x^{2} – x^{4})^{1/2}.

**Solution**: The given function represents a right side of a loop which intersects the x-axis at (0, 0) and (2, 0). Now the area of the reqired region will be

**Example**: Find the area bounded by closed curve given by the equation |y| = 1 – x^{2}.

**Solution**: The given function represents two parabola y = x^{2} – 1 and y = 1 – x^{2} with vertices (0, -1) and (0, 1). Both parabola together forms a closed curve and intesect the x-axis at (-1, 0) and (1, 0). Given functionscollectively will represent a shape similar to a circle (fig. 2.1 in thumbnail). Hence the required area of the desired region will be

\begin{aligned}Area=4\int ^{1}_{0}\left( 1-x^{2}\right) dx=4\int ^{1}_{0}dx-4\int ^{1}_{0}x^{2}dx\\ \Rightarrow 4-\dfrac{4}{3}\left( x^{3}\right) _{0}^{1}=4-\dfrac{4}{3}=\dfrac{8}{3}\end{aligned}

**Example**: Find the area of the closed region bounded by the curve given by the equation (y – Sin^{-1}x)^{2} = x – x^{2}.

**Solution**: Consider the figure 3.1 given in the thumbnail. Since x – x^{2} ≥ 0, 0 ≤ x ≤ 1 and y = Sin^{-1}x ± (x – x^{2})^{1/2}. Hence the given equation represents two branches of the given function. Thease branches intersect at (0, 0) and (1, π/2). Hence the required area will be given by

\begin{aligned}\int ^{1}_{0}\left( \left( \sin ^{-1}\left( x\right) +\sqrt{x-x^{2}}\right) -\left( \sin ^{-1}\left( x\right) -\sqrt{x-x^{2}}\right) \right) dx\\ =2\int ^{1}_{0}\left( \sqrt{x-x^{2}}\right) dx=\dfrac{1}{4}\left( \dfrac{\pi }{2}+\dfrac{\pi }{2}\right) \\ \ =\dfrac{\pi }{4}\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 Of The Region Bounded By the Curves Given In their Parametric Form”**.