# Finding Area Bounded by Two Curves Using Definite Integrals

### Finding Area Bounded by Two Curves Using Definite Integrals:

1. Objective:

• Calculate the area between two curves $y=f\left(x\right)$ and $y=g\left(x\right)$ over the interval $\left[a,b\right]$ using definite integrals.
2. Approach:

• The area between two curves on the interval $\left[a,b\right]$ is the definite integral of the difference between the top curve $f\left(x\right)$ and the bottom curve $g\left(x\right)$: $\text{Area}={\int }_{a}^{b}\left(f\left(x\right)-g\left(x\right)\right)\text{\hspace{0.17em}}dx$
• This method calculates the vertical distance between the curves and integrates over the interval to find the area.
3. Example: Consider the curves $y={x}^{2}$ and $y=x$ over the interval $\left[0,1\right]$. Let's find the area enclosed between these curves.

4. Steps to Find Area:

• Step 1: Identify Enclosed Region: Determine the curves $y={x}^{2}$ and $y=x$ and the interval $\left[0,1\right]$ enclosing the desired area.

• Step 2: Determine Integration Limits: Find the intersection points of the curves to define the bounds for the definite integral.

• Solve ${x}^{2}=x$ to find the intersection points: ${x}^{2}-x=0$
• Factoring: $x\left(x-1\right)=0$, yielding $x=0$ and $x=1$.
• Step 3: Setup Definite Integral: The area between $y={x}^{2}$ and $y=x$ over $\left[0,1\right]$ is given by: $\text{Area}={\int }_{0}^{1}\left(x-{x}^{2}\right)\text{\hspace{0.17em}}dx$

• Step 4: Evaluate Integral: Compute the definite integral: $\text{Area}={\left[\frac{{x}^{2}}{2}-\frac{{x}^{3}}{3}\right]}_{0}^{1}=\left[\left(\frac{1}{2}-\frac{1}{3}\right)-\left(0-0\right)\right]=\left[\frac{1}{6}\right]=\frac{1}{6}$

5. Result:

• The area enclosed between the curves $y={x}^{2}$ and $y=x$ over the interval $\left[0,1\right]$ is $\frac{1}{6}$ square units.