Applications of Integration: Area Between Curves and Related Concepts

Applications of Integration

Area Between Curves

The area between two curves is a fundamental application of definite integrals in calculus. When two functions bound a region on a given interval, the area can be found by integrating the difference of the functions over that interval.

• Definition: If f(x) and g(x) are continuous functions 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:

$A = \int_a^b [f(x) - g(x)]\,dx$

• Key Steps:

  1. Find the points of intersection of the curves to determine the limits of integration.

  2. Identify which function is on top (greater value) and which is on the bottom (lesser value) over the interval.

  3. Set up the integral of the top function minus the bottom function over the interval.

• Example: Find the area between y = 2x - x^2 and y = x.

  • Set 2x - x^2 = x to find intersection points: x^2 - x = 0 \implies x(x-1) = 0 \implies x = 0, 1.

  • On [0,1], 2x - x^2 \geq x.

  • Area: $A = \int_0^1 [(2x - x^2) - x] dx = \int_0^1 (x - x^2) dx$

  • Compute: $\int_0^1 (x - x^2) dx = \left[\frac{1}{2}x^2 - \frac{1}{3}x^3\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

Area with Respect to y (Horizontal Slices)

Sometimes, it is more convenient to integrate with respect to y, especially when the functions are given as x in terms of y.

• Formula: If x = f(y) is to the right of x = g(y) on [c, d], then:

$A = \int_c^d [f(y) - g(y)]\,dy$

• Example: Find the area between x = y^2 and x = y + 2 for y in [a, b].

Area Enclosed by Intersecting Curves

When two curves intersect, the area between them is found by integrating between their points of intersection.

• Key Steps:

  1. Set the equations equal to find intersection points.

  2. Determine which function is on top/bottom (or right/left for y-integration).

  3. Integrate the difference over the interval.

• Example: Area between y = \sin x and y = \cos x from x = 0 to x = \frac{\pi}{4}.

  • Area: $A = \int_0^{\frac{\pi}{4}} (\cos x - \sin x) dx$

  • Compute: $\int (\cos x - \sin x) dx = \sin x + \cos x + C$

  • Evaluate: $[\sin x + \cos x]_0^{\frac{\pi}{4}} = (\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}) - (0 + 1) = \sqrt{2} - 1$

Special Cases: Area Between Parabolas

When the region is bounded by two parabolas, the process is the same: find intersection points, determine which is on top, and integrate the difference.

• Example: Area between y = 2x - x^2 and y = x^2.

  • Set 2x - x^2 = x^2 to find intersection: 2x - 2x^2 = 0 \implies x(1 - x) = 0 \implies x = 0, 1.

  • Area: $A = \int_0^1 [(2x - x^2) - x^2] dx = \int_0^1 (2x - 2x^2) dx$

  • Compute: $\int_0^1 (2x - 2x^2) dx = [x^2 - \frac{2}{3}x^3]_0^1 = 1 - \frac{2}{3} = \frac{1}{3}$

Summary Table: Area Between Curves

Situation	Formula	Notes
Vertical Slices (y as function of x)	$A = \int_a^b [f(x) - g(x)] dx$	f(x) is above g(x) on [a, b]
Horizontal Slices (x as function of y)	$A = \int_c^d [f(y) - g(y)] dy$	f(y) is to the right of g(y) on [c, d]
Enclosed by intersection	Integrate between intersection points	Find limits by solving f(x) = g(x)

Additional info:

• Some notes referenced the Comparison Test and Convergence of Series, but the main focus is on area between curves.

• For more complex regions, it may be necessary to split the integral at points where the top/bottom function changes.

• Applications include finding the area of regions bounded by trigonometric, exponential, or logarithmic functions.