Definite Integrals, Riemann Sums, and Calculating Areas Bounded by Curves

Approximations, Estimation, and Riemann Sum Foundations

• Estimations can be checked for logical plausibility even when dealing with non-linear curves rather than straight lines.
  • For example, approximating an integral value of $\frac{20}{3}$ can be validated by comparing it to $\frac{21}{3} = 7$. Since $\frac{21}{3} = 7$, a value around $7$ logically matches expectations for $\frac{20}{3}$.
• Riemann sums estimate the area under a curve by partitioning an interval into $n$ sub-intervals or slices, each having a uniform width represented by $\Delta x$.
• To calculate the area of an individual rectangular slice, the height of the rectangle is evaluated using the function value $f(x_i)$ at a specific sample point within the sub-interval:
  • Left Riemann Sum: Evaluates the function height at the left endpoint of each sub-interval.
  • Right Riemann Sum: Evaluates the function height at the right endpoint of each sub-interval.
  • Midpoint Riemann Sum: Evaluates the function height at the midpoint of each sub-interval.
  • General Sample Point: Evaluates height at any arbitrarily selected point $x_i$ within the $i$-th sub-interval.
• The area of a single rectangular slice is given by the height multiplied by the width:   $A_{\text{slice}} = f(x_i) \Delta x$
• The total area estimation across all $n$ rectangular slices is expressed using summation notation:   $\text{Estimated Area} = \sum_{i=1}^{n} f(x_i) \Delta x$

Definite Integrals and Area Relative to the X-Axis

• As the number of sub-intervals $n$ approaches infinity ($n \rightarrow \infty$), the width $\Delta x$ approaches zero, making the area estimation exact.
• At the limit $n \rightarrow \infty$, the Left Riemann Sum and Right Riemann Sum converge to the exact same value, which defines the definite integral:   $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i) \Delta x = \int_a^b f(x)\,dx$
• Relationship between definite integrals and geometric area:
  • A definite integral equals the exact geometric area enclosed between a curve $y = f(x)$ and the x-axis if and only if $f(x) \ge 0$ (the function lies entirely on or above the x-axis) across the entire interval $[a, b]$.
  • When $f(x) \ge 0$, every evaluated function height $f(x_i)$ is positive, ensuring $f(x_i) \Delta x > 0$
  • When $f(x) < 0$ (the function lies below the x-axis), the evaluated function values $f(x_i)$ are negative, yielding negative products $f(x_i) \Delta x < 0$
  • Summing these negative rectangular terms results in subtraction rather than addition, causing the definite integral to measure net signed area rather than total geometric enclosed area.

Discrepancy Between Net Signed Area and Total Enclosed Area

• Demonstration using the function $y = x^2 - 1$:
  • Consider evaluating the area bounded by $y = x^2 - 1$ and the x-axis.
  • Naively computing the definite integral from $x = 0$ to $x = 3$ gives:     $\int_0^3 (x^2 - 1)\,dx = 6$
  • However, the actual total geometric area enclosed across this interval is $\frac{20}{3}$.
  • Reason for the discrepancy:
  • The function $y = x^2 - 1$ drops below the x-axis on the interval $[0, 1]$ and rises above the x-axis on the interval $[1, 3]$.
  • On $[0, 1]$, $f(x_i)$ is negative, producing a negative area evaluation $A_1$ below the x-axis.
  • On $[1, 3]$, $f(x_i)$ is positive, producing a positive area evaluation $A_2$ above the x-axis.
  • The naive integral calculates $A_2 - A_1 = 6$, subtracting the lower area from the upper area.
• Correct procedure for finding total enclosed geometric area:
  • To obtain the true total enclosed area ($A_2 + A_1$), the region must be partitioned at the x-intercepts (where $f(x) = 0$).
  • For $y = x^2 - 1$, setting $x^2 - 1 = 0$ yields the x-intercept at $x = 1$
  • Separate the integral into sub-regions above and below the x-axis:
  • Region below the x-axis ($[0, 1]$): Negate the integral to convert the negative value into a positive geometric area:       $-\int_0^1 (x^2 - 1)\,dx$
  • Region above the x-axis ($[1, 3]$): Integrate directly:       $\int_1^3 (x^2 - 1)\,dx$
  • Sum the positive components to calculate total enclosed area:     $\text{Total Enclosed Area} = -\int_0^1 (x^2 - 1)\,dx + \int_1^3 (x^2 - 1)\,dx = \frac{20}{3}$
• Explanation of integral output $6$:
  • The calculated integral value of $6$ represents the net signed area, where the area below the x-axis ($A_1 = 1$ or $\frac{2}{3}$) was subtracted from the area above the x-axis ($A_2 = 7$ or $\frac{22}{3}$, yielding $6$).

Calculating Area Bounded Between Curves

• Generalizing area calculations to regions bounded between two or more curves:
  • Step 1: Always graph the functions first to visually identify spatial configurations, upper and lower boundaries, and points of intersection.
  • Example curve configuration: Parabola $y = x^2$ with its vertex located at $(0, 0)$.
  • For two functions where $f(x)$ lies completely above $g(x)$ on an interval $[a, b]$:
  • Top function: $y_{\text{top}} = f(x)$
  • Bottom function: $y_{\text{bottom}} = g(x)$
  • Area under the top function $f(x)$ above the x-axis is $\int_a^b f(x)\,dx$
  • Area under the bottom function $g(x)$ above the x-axis is $\int_a^b g(x)\,dx$
  • Subtracting the lower region from the upper region isolates the bounded area between the curves:       $\text{Bounded Area} = \int_a^b f(x)\,dx - \int_a^b g(x)\,dx = \int_a^b (f(x) - g(x))\,dx$

Generalization of the Top-Minus-Bottom Rule

• Proof that relative position to the x-axis does not affect the formula:
  • Case where $f(x)$ (top function) is above the x-axis and $g(x)$ (bottom function) is below the x-axis:
  • Let $A_1$ be the geometric area under $f(x)$ above the x-axis: $\int_a^b f(x)\,dx = A_1$
  • Let $A_2$ be the geometric area bounded by $g(x)$ below the x-axis: $\int_a^b g(x)\,dx = -A_2$
  • Applying the formula $\int_a^b (f(x) - g(x))\,dx$ gives:       $\int_a^b f(x)\,dx - \int_a^b g(x)\,dx = A_1 - (-A_2) = A_1 + A_2$
  • The expression automatically converts the negative integral into a positive addition, yielding the exact total enclosed geometric area between the two curves.
• General Rule for Bounded Area:
  • Regardless of whether functions sit above, below, or cross the x-axis, the enclosed area $A$ between an upper curve $y_{\text{top}}$ and a lower curve $y_{\text{bottom}}$ from $x = a$ to $x = b$ is always:     $A = \int_a^b (y_{\text{top}} - y_{\text{bottom}})\,dx$
• Unification with single-variable x-axis integration:
  • When $f(x) \ge 0$, the x-axis is represented by the line $y = 0$ acting as $y_{\text{bottom}}$:     $A = \int_a^b (f(x) - 0)\,dx = \int_a^b f(x)\,dx$
  • When $f(x) \le 0$, the x-axis $y = 0$ acts as $y_{\text{top}}$ and $f(x)$ acts as $y_{\text{bottom}}$:     $A = \int_a^b (0 - f(x))\,dx = -\int_a^b f(x)\,dx$
• Key Procedural Requirement:
  • Graphing the functions is an indispensable first step to explicitly verify which function is $y_{\text{top}}$ and which is $y_{\text{bottom}}$, as well as determining any required x-bounds or intersection points.