Working Rules for Finding Area Using Definite Integrals

Working Rules for Finding Area Using Definite Integrals:

  1. Positive Function:

    • Rule 1: For a non-negative function f(x)0 over an interval [a,b], the area under the curve y=f(x) from x=a to x=b is given by the definite integral: Area=abf(x)dx
    • This rule applies when the function remains above the x-axis for the entire interval, representing the area between the curve and the x-axis.
  2. Multiple Regions and Piecewise Functions:

    • Rule 2: If the function f(x) changes sign or has multiple regions over [a,b], find the areas of each region separately and sum their absolute values to get the total area. Total Area=acf(x)dx+cbf(x)dx
    • Here, c is the point where f(x) changes sign or the regions switch.
  • Area between Curves:

    • Rule 3: When finding the area between two curves f(x) and g(x) over [a,b], compute the definite integral of their absolute difference: Area=abf(x)g(x)dx
    • This rule applies to determine the area enclosed by both curves within the given interval.
  • Negative Function:

    • Rule 4: If f(x)0 over [a,b], the definite integral abf(x)dx gives the negative of the area between the curve y=f(x) and the x-axis.
      • The magnitude of this integral represents the area, but the negative sign indicates that the area lies below the x-axis.
  • Visualization and Graphical Analysis:

    • Rule 5: Graphical representations help visualize the regions and understand the behavior of functions with respect to the x-axis for accurate area calculation using definite integrals.
    • Riemann sums or numerical approximations aid in understanding and approximating areas under curves.

Example:

Consider the function f(x)=x22x over the interval [0,2]. Let's find the area between the curve y=f(x) and the x-axis within this interval.

  1. Identify the Region:

    • The function f(x)=x22x intersects the x-axis at x=0 and x=2.
    • To find the area between the curve and the x-axis, we'll compute the integral 02f(x)dx.
  2. Integral Calculation: Area=02x22xdx

  • Factorizing: x22x=x(x2)
  • Over [0,2], x(x2) is positive, so x22x=x(x2)
  • Integral Setup and Solution: Area=02x(x2)dx

    • Compute the integral: Area=[x33x2]02=[23322][03302]Area=[834][00]=834=8123=43
  • Result:

    • The area between the curve y=x22x and the x-axis over [0,2] is 43 square units.
    • The negative sign indicates that this area lies below the x-axis due to the nature of the function.