Applications of Definite Integrals

Computing Areas Between Curves

Definite integrals can be used to compute the area of regions bounded by two curves. This is a fundamental application in calculus, extending the concept of finding the area under a single curve to the area between two curves. The method involves summing the areas of thin rectangular slices, which can be oriented either vertically or horizontally, depending on the functions involved.

Area Using Vertical Slices (Integration with Respect to x)

Definition and Setup

When the region is bounded by curves expressed as y = f(x) (upper curve) and y = g(x) (lower curve), and vertical lines x = a and x = b, the area between the curves is found by integrating with respect to x. The functions f and g must be continuous on the interval [a, b], and f(x) ≥ g(x) for all x in [a, b].

• Region S:

• Approximating the Area: Divide [a, b] into n subintervals of equal width , and approximate the area using rectangles.

• Height of Rectangle:

• Area of Rectangle:

• Riemann Sum:

• Exact Area (as n → ∞):

Algebraic Justification

The area under y = f(x) from x = a to x = b is . The area under y = g(x) over the same interval is . The area between the curves is the difference:

Example 1: Area Between Exponential and Linear Functions

Find the area of the region bounded above by y = e^x, below by y = x, and on the sides by x = 0 and x = 1.

• Setup:

• Evaluation:

Example 2: Area Between Two Parabolas

Find the area of the region enclosed by y = x^2 and y = 2x - x^2.

• Intersection Points: Solve or

• Upper Curve:

• Lower Curve:

• Integral:

• Evaluation:

Example 3: Area Between Sine and Cosine

Find the area of the region bounded by y = \sin x, y = \cos x, x = 0, and x = \frac{\pi}{2}.

• Intersection:

• For , ; for ,

• Integral:

• Evaluation:

Area Using Horizontal Slices (Integration with Respect to y)

Definition and Setup

When the region is bounded by curves expressed as x = f(y) (right curve) and x = g(y) (left curve), and horizontal lines y = c and y = d, the area is found by integrating with respect to y. The functions f and g must be continuous on [c, d], and f(y) ≥ g(y) for all y in [c, d].

• Area Formula:

• Interpretation: Area = (right curve) − (left curve)

Example: Area Enclosed by a Line and a Parabola

Find the area enclosed by the line y = x - 1 and the parabola y^2 = 2x + 6.

• Intersection Points: (−1, −2) and (5, 4)

• Right Boundary:

• Left Boundary:

• Integral:

• Evaluation: