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

Objective:
 Calculate the area between two curves $y=f(x)$and $y=g(x)$over the interval $[a,b]$ using definite integrals.

Approach:
 The area between two curves on the interval $[a,b]$ is the definite integral of the difference between the top curve $f(x)$ and the bottom curve $g(x)$: $\text{Area}={\int}_{a}^{b}(f(x)g(x))\text{\hspace{0.17em}}dx$
 This method calculates the vertical distance between the curves and integrates over the interval to find the area.

Example: Consider the curves $y={x}^{2}$ and $y=x$ over the interval $[0,1]$. Let's find the area enclosed between these curves.

Steps to Find Area:

Step 1: Identify Enclosed Region: Determine the curves $y={x}^{2}$ and $y=x$ and the interval $[0,1]$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: $x(x1)=0$, yielding $x=0$ and $x=1$.

Step 3: Setup Definite Integral: The area between $y={x}^{2}$ and $y=x$ over $[0,1]$ is given by: $\text{Area}={\int}_{0}^{1}(x{x}^{2})\text{\hspace{0.17em}}dx$

Step 4: Evaluate Integral: Compute the definite integral: $\text{Area}={[\frac{{x}^{2}}{2}\frac{{x}^{3}}{3}]}_{0}^{1}=[(\frac{1}{2}\frac{1}{3})(00)]=\left[\frac{1}{6}\right]=\frac{1}{6}$


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