Areas Between Curves: Fresh Take

  • Calculate the area between two curves by integrating with respect to [latex]x[/latex]
  • Calculate the area of a compound region
  • Calculate the area between two curves by integrating with respect to [latex]y[/latex]
  • Determine the most effective variable, [latex]x[/latex] or [latex]y[/latex], for integration based on the curves’ orientation

Area of a Region between Two Curves

The Main Idea 

  • Extends the concept of definite integrals to calculate areas of more complex regions
  • For continuous functions [latex]f(x)[/latex] and [latex]g(x)[/latex] where f(x) ≥ g(x) on [latex][a,b][/latex]:
    • [latex]A = \int_a^b [f(x) - g(x)] dx[/latex]
  • Area represents the difference between the upper and lower function, integrated over the interval
  • Process:
    • Identify the upper and lower functions
    • Determine the interval of integration
    • Set up and evaluate the definite integral
  • Often necessary to solve [latex]f(x) = g(x)[/latex] to determine integration limits

If [latex]R[/latex] is the region bounded by the graphs of the functions [latex]f(x)=\frac{x}{2}+5[/latex] and [latex]g(x)=x+\frac{1}{2}[/latex] over the interval [latex]\left[1,5\right],[/latex] find the area of region [latex]R.[/latex]

If [latex]R[/latex] is the region bounded above by the graph of the function [latex]f(x)=x[/latex] and below by the graph of the function [latex]g(x)={x}^{4},[/latex] find the area of region [latex]R.[/latex]

Areas of Compound Regions

The Main Idea 

  • Extends area calculations to regions where functions intersect
  • Key formula for regions bounded by intersecting curves [latex]f(x)[/latex] and [latex]g(x)[/latex] on [latex][a,b][/latex]:
    • [latex]A = \int_a^b |f(x) - g(x)| dx[/latex]
  • Absolute value in the formula accounts for changes in which function is greater
  • Process:
    • Identify intersection points of the functions
    • Break the interval into subintervals where one function is consistently greater
    • Set up and evaluate separate integrals for each subinterval
    • Sum the results to get the total area

If [latex]R[/latex] is the region between the graphs of the functions [latex]f(x)= \sin x[/latex] and [latex]g(x)= \cos x[/latex] over the interval [latex]\left[\frac{\pi}{2},2\pi \right],[/latex] find the area of region [latex]R.[/latex]

Consider the region depicted below. Find the area of [latex]R.[/latex]

This figure is has two graphs in the first quadrant. They are the functions f(x) = x^2 and g(x)= 2-x. In between these graphs is a shaded region, bounded to the left by f(x) and to the right by g(x). All of which is above the x-axis. The region is labeled R. The shaded area is between x=0 and x=2.
Figure 6. Two integrals are required to calculate the area of this region.

Find the area of the region bounded by [latex]y = x^2[/latex] and [latex]y = 4 - x^2[/latex].

Regions Defined with Respect to [latex]y[/latex]

The Main Idea 

  • Alternative approach to finding areas between curves by integrating with respect to [latex]y[/latex]
  • Useful when curves are more easily expressed as functions of [latex]y[/latex]
  • Can simplify calculations by requiring only one integral instead of two
  • Key formula: [latex]A = \int_c^d [u(y) - v(y)] dy[/latex], where [latex]u(y)[/latex] is the right boundary and [latex]v(y)[/latex] is the left boundary
  • Process:
    • Express curves as functions of [latex]y[/latex]
    • Determine the limits of integration ([latex]y[/latex]-coordinates of intersection points)
    • Set up and evaluate the integral

Find the area of the region bounded by [latex]y = x^2[/latex] and [latex]y = 2x[/latex].

Find the area of the region in the first quadrant bounded by [latex]x^2 + y^2 = 4[/latex] and [latex]y = x[/latex].