Applications of Integration - Areas Between Curves
6 Applications of Integration
This section elaborates on the applications of integration focusing on calculating the area between curves, providing thorough explanations, examples, and formulas used for this process.
6.1 Areas Between Curves
Introduction
The concept of finding areas between curves is crucial in calculus, particularly when using integration techniques.
This section focuses on two methods of integration:
With respect to x
With respect to y
Area Between Curves: Integrating With Respect to x
Definition of the Area Between Curves: Given the region S, defined between two curves:
$y = f(x)$
$y = g(x)$
The axes are bounded by vertical lines $x = a$ and $x = b$ where $f$ and $g$ are continuous functions such that $f(x) ≥ g(x)$ for all $x$ in the interval $[a, b]$.
Visualization: Refer to Figure 1 for a graphical representation.
Approximation of Area Using Rectangles
To approximate the area of the region S, divide it into n strips of equal width ($Δx$).
Each strip's height will be approximated by the difference between the two functions at a sample point $x_i^*$:
Height = $f(xi^) - g(xi^)$
Riemann Sum: The sum approximating the area (i.e., the sum of the areas of rectangles) converges to the actual area as $n → ∞$:
$A ≈ ext{lim}{n o ext{∞}} ext{sum}(f(xi^) - g(x_i^))$
This leads to the definition of area A of the region S:
$A = ext{lim}_{n o ext{∞}} ext{Riemann Sum} = ext{definite integral of } (f - g)$
Area Formula
The area A of the region S is given by the integral:
A = \int_{a}^{b} [f(x) - g(x)] \, dx
When $g(x) = 0$, the area formula reduces to the area under the graph of $f(x)$.
In scenarios where both functions $f$ and $g$ are positive, the area can be conceptualized as:
A = ext{[area under } y = f(x)] - ext{[area under } y = g(x)]
Refer to Figure 3 for visualization.
Example 1: Area Calculation
Problem Description: Find the area of the region bounded above by $y = f(x)$ and below by $y = g(x)$, with lateral boundaries $x = 0$ and $x = 1$.
Solution Method:
Use the area formula with defining limits (a = 0, b = 1).
Sketch the region, identifying top curve (yT) and bottom curve (yB) for appropriate approximating rectangles (Figure 4 and Figure 5).
Dealing with Multiple Regions
When $f(x) ≥ g(x)$ holds for some values of x but $g(x) ≥ f(x)$ for others, the region can be divided into smaller regions $S1, S2, ext{…}$:
The overall area $A$ will equal the sum of the areas of the smaller regions $A = A1 + A2 + …$.
Integral representation:
A = \int_{a}^{b} [f(x) - g(x)] \, dx for regions where layers need to be integrated separately.
Refer to Figure 8 for clarification.
Example 4: Area Between Trigonometric Functions
Problem Statement: Calculate the area bounded by $y = ext{sin}(x)$, $y = ext{cos}(x)$, $x = 0$, and the intersection points.
Finding Points of Intersection: Set $ ext{sin}(x) = ext{cos}(x)$.
Use the area formula structured as:
A = \int{0}^{\frac{π}{4}} ( ext{cos}(x) - ext{sin}(x)) \, dx + \int{\frac{π}{4}}^{π/2} ( ext{sin}(x) - ext{cos}(x)) \, dx
Each part needs evaluation based on limits defined by intersection points derived earlier (Reference Figure 9).
Area Between Curves: Integrating With Respect to y
Some regions are more manageable when considered with respect to y. Given curves defined by:
$x = f(y)$ and $x = g(y)$ bounded by $y = c$ and $y = d$, the area can be calculated similarly:
A = \int_{c}^{d} [ f(y) - g(y)] \, dy
Visualization in Figure 10 exemplifies how x coordinates change in accordance with y boundaries.
Example 5: Specific Region Calculation
Given Curves: The line $y = x - 1$ and the parabola defined.
Intersection Points: Solve $x - 1 = ext{parabola equation}$ to find limits for area computation.
Integral Evaluation: Compute as:
A = \int_{-2}^{4} [R(c) - L(c)] \, dy for relevant boundaries as extracted from intersection points (Figure 12).
Applications Beyond Geometry
Velocity Curves: The area between two velocity curves represents the distance between two objects after a specific interval of time.
Example 7: Two cars A and B's distance after their respective accelerations is calculated by using the Midpoint Rule to estimate the area under the velocity curves.
Velocity Data for Cars: Tabulated velocities at certain intervals with numerical conversions to feet per second.
The integration of distance calculations is expressed as:
ext{Distance between cars} = 4 imes ext{average velocity difference} = 372 ext{ feet}