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(x) and y=g(x) over the interval [a,b] using definite integrals.
  2. 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): Area=ab(f(x)g(x))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=x2 and y=x over the interval [0,1]. Let's find the area enclosed between these curves.

  4. Steps to Find Area:

    • Step 1: Identify Enclosed Region: Determine the curves y=x2 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 x2=x to find the intersection points: x2x=0
      • Factoring: x(x1)=0, yielding x=0 and x=1.
    • Step 3: Setup Definite Integral: The area between y=x2 and y=x over [0,1] is given by: Area=01(xx2)dx

    • Step 4: Evaluate Integral: Compute the definite integral: Area=[x22x33]01=[(1213)(00)]=[16]=16

  5. Result:

    • The area enclosed between the curves y=x2 and y=x over the interval [0,1] is 16 square units.