Applications of Integration

Regions Between Curves

In calculus, the area between two curves can be found using definite integrals. This technique generalizes the method for finding the area under a single curve and is essential for solving a variety of applied and theoretical problems.

Definition and Basic Principle

• Area Between Two Curves (with respect to x): If f(x) and g(x) are continuous on [a, b] and f(x) \geq g(x) for all x in [a, b], then the area A between the curves from x = a to x = b is given by:

• Slice-and-Sum Strategy: The region is divided into thin vertical slices (rectangles), and the area is approximated by summing the areas of these rectangles. As the number of slices increases, the approximation approaches the exact area, represented by the definite integral.

• Special Case: If the lower curve is the x-axis (i.e., g(x) = 0), the formula reduces to the standard area under y = f(x).

Example: Area Between Two Parabolas

• Problem: Find the area between f(x) = 5 - x^2 and g(x) = x^2 - 3 from x = -2 to x = 2.

• Solution: The area is:

• Evaluate the integral to find the exact area.

Compound Regions

• When the region is bounded by more than two curves or the upper/lower curve changes, split the region at intersection points and sum the areas of the subregions.

Integrating with Respect to y

Sometimes, it is more convenient to integrate with respect to y, especially when the curves are functions of y (i.e., x = f(y), x = g(y)).

• Area Between Two Curves (with respect to y): If f(y) \geq g(y) on [c, d], then:

• Here, the region is sliced horizontally, and the area is the sum of the lengths of the horizontal slices.

Example: Area with Respect to y

• Problem: Find the area of the region bounded by x = y + 4 and x = y^2 from y = 0 to y = 2.

• Solution:

Alternative Forms and Properties

• The area between two curves can also be expressed as the difference of two definite integrals:

• This is useful when each curve represents a physical quantity, such as velocity or cost.

Examples of Regions Between Curves

• Below are several common examples of regions between curves, each with a shaded area representing the region whose area is to be calculated using definite integrals.

Applications and Interpretation

• Physical Meaning: The area between two curves can represent quantities such as the difference in distance traveled, net profit, or the probability between two distributions.

• Compound and Complicated Regions: For regions bounded by more than two curves or with changing boundaries, divide the region into simpler subregions and sum their areas.

Summary Table: Area Formulas

Additional info: The images included above are directly relevant to the explanation of regions between curves and illustrate the geometric interpretation of the definite integral as area. They show various configurations of curves and the corresponding shaded regions whose area is to be computed.