6.2 Regions Between Curves

Introduction to Regions Between Curves

The area between two curves is a fundamental application of definite integrals in calculus. This topic involves finding the area of a region bounded by two or more functions, typically using integration. The process can be performed with respect to either the x-axis or the y-axis, depending on the orientation of the curves.

Finding Area Between Curves: The Slice-and-Sum Strategy

To find the area between two curves, we use the slice-and-sum strategy, which involves slicing the region into thin rectangles, summing their areas, and taking the limit as the number of slices approaches infinity. This leads to the definite integral formula for the area.

• Key Formula (with respect to x):

• f(x): The upper function (top curve)

• g(x): The lower function (bottom curve)

• [a, b]: The interval over which the region is bounded

Example 1: Area Between Two Parabolas

Consider the region bounded by and from to .

• Set up the integral:

• Evaluate the integral:

Example 2: Area Between a Line and a Cubic

Find the area of the region bounded by , , and .

• Set up the integral (split at ):

• Evaluate:

Integrating with Respect to y

Sometimes, it is more convenient to integrate with respect to y, especially when the region is bounded by functions of y or when vertical slices are not possible. In this case, the area is given by:

• f(y): The rightmost function (in terms of x)

• g(y): The leftmost function (in terms of x)

• [c, d]: The interval over which the region is bounded in y

Example 3: Area with Respect to y

Find the area of the region bounded by and from to .

• Set up the integral:

• Evaluate:

Step-by-Step Example: Finding Area Between Curves

1. Find the intersection points (boundaries): Set the two functions equal and solve for the variable.

2. Set up the definite integral: Integrate the difference (top minus bottom or right minus left) over the interval.

3. Evaluate the integral: Compute the definite integral to find the area.

Example: Find the area between and .

• Find intersection points:

• Set up the integral:

• Evaluate:

Advanced Example: Area Between Curves in Terms of y

Find the area of the region bounded by and .

• Find intersection points:

• Solve for y:

• Set up the integral:

• Evaluate:

Summary Table: Steps for Finding Area Between Curves

Additional info:

• When the region is bounded by more than two curves, or the top/bottom (or right/left) function changes, split the region into subregions and sum the areas.

• Always sketch the region to determine which function is on top/bottom or right/left.