Area Calculation in Integrals

In advanced calculus, comprehending how to calculate areas under curves is crucial. This guide delves into key concepts, techniques, and real-world applications involving areas in integral calculus.

Key Term: Area under a Curve

• Importance: A core principle in integral calculus, utilised to compute cumulative values over intervals.
• Applications: Employed in various fields, such as physics (displacement) and economics (cost analysis).

Definite Integrals and Their Context

Real-world Applications

• Engineering: Determine plot areas for construction.
• Physics: Calculate total distance using velocity functions.
• Economics: Assess consumer surplus beneath demand curves.

Understanding these applications underscores the importance of integrals.

Definite Integral Notation

A definite integral, symbolised as $\int_a^b f(x) \, dx$, encompasses:

• $\int$: The integration process.
• $a, b$: The limits of integration.
• $f(x)$: The function to integrate.
• $dx$: Specifies integration concerning $x$.

Step-by-Step Calculation of Area

1. Identify the function $f(x)$.
2. Set Limits from $a$ to $b$.
3. Evaluate the definite integral.

Examples:

• Linear Example: Integrate $f(x) = x$ from $0$ to $2$; $\int_0^2 x \, dx = 2$.

• Polynomial Example: For $f(x) = x^2$ from $1$ to $3$, compute by $\left[ \frac{x^3}{3} \right]_1^3$.

Signed Areas and Their Calculation

• Signed Areas: Denotes the total area under a curve, where the area above the x-axis is positive and below is negative.

Calculation Example

1. Determine Anti-derivative: For $f(x) = x - 1$, $F(x) = \frac{x^2}{2} - x$.
2. Apply Limits: $\int_0^2 (x - 1) \, dx = F(2) - F(0) = 0$.

Approximation Techniques

Introduction

• Essential when exact integration proves challenging due to complexities involving non-linear or transcendental functions.

Methods

Rectangular Methods

• Riemann Sums: Utilise rectangles to estimate areas.
  • Left/Right/Midpoint Riemann Sums offer varying measurement points.

Trapezoidal Rule

• Improves rectangle approximations by employing trapezoids.
• Useful for complex functions lacking straightforward anti-derivatives.

$A \approx \frac{b-a}{2n} \sum_{i=1}^{n}[f(x_{i-1}) + f(x_i)]$

• Example: Determine the area under $f(x) = x^2$ on $[0,2]$, $n=4$ using the trapezoidal method.

Example: Trapezoidal Rule with $f(x) = x^2$

1. Choose Intervals: Consider lines $[0, 1]$ and $[1, 2]$.
2. Evaluate at endpoints; accumulate areas.

For example, with $n=2$ (dividing $[0,2]$ into two equal subintervals):

• Endpoints are $x_0=0$, $x_1=1$, $x_2=2$
• Function values are $f(0)=0$, $f(1)=1$, $f(2)=4$
• Using the formula: $A \approx \frac{2-0}{2(2)}[(0+1) + (1+4)] = \frac{1}{2}[1 + 5] = 3$
• Exact value is $\frac{8}{3}$, so our approximation is reasonably close.

Symmetry in Calculations

• Symmetry in Functions: Simplifies calculations when working with even ($f(x)=f(-x)$) and odd functions ($f(x)=-f(-x)$).
• Application: Reduces redundant integrations when symmetry is applicable.

Example

• $\int_{-b}^{b} x^2 \, dx = 2 \int_{0}^{b} x^2 \, dx$ helps simplify calculations.

Fundamental Theorem of Calculus (FTC)

Explanation

• Connection between differentiation and integration.
• Formula: $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$
  • F is an antiderivative of f.

Example

• Polynomial: To evaluate $\int_0^3 (2x) \, dx$, use $F(x) = x^2$. Solution: $F(3) - F(0) = 3^2 - 0^2 = 9$

• Trigonometric: For $\int_0^{\pi} \sin(x) \, dx$, apply $F(x) = -\cos(x)$. Solution: $-\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2$

Exam Tips

• Always verify that limits are arranged with $a < b$.
• Use symmetry to streamline calculations.

Grasping these concepts and methods will turn integral calculus into a valuable tool for exams and practical applications.