2.3 Differential and Integral Calculus

2.3 Differential and Integral Calculus

Differential and integral calculus provides the essential mathematical engine for civil engineering design, governing beam deflections, fluid kinematics, geotechnical settlement rates, and structural cross-sectional properties. In the CELE examination, calculus problems require rapid differentiation, applied optimization, multi-variable related rates, integration techniques, and the calculation of plane areas, volumes of revolution, and centroids.

---

Limits, Continuity & L'Hôpital's Rule

Indeterminate Forms

When evaluating the limit $\lim_{x \to c} \frac{f(x)}{g(x)}$ yields indeterminate ratios of type $\left[\frac{0}{0}\right]$ or $\left[\frac{\infty}{\infty}\right]$, L'Hôpital's Rule establishes that: $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ Provided the derivatives exist and the resulting limit converges or diverges to infinity. (Caution: Do not apply the quotient rule when using L'Hôpital's Rule; differentiate the numerator and denominator independently!)

Standard Fundamental Limits

• $\lim_{x \to 0} \frac{\sin x}{x} = 1 \quad (x \text{ in radians})$
• $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$
• $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \approx 2.71828$

---

Differential Calculus & Optimization

Core Differentiation Rules

• Product Rule: $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$
• Quotient Rule: $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$
• Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
• Transcendental Derivatives: $\frac{d}{dx}(\sin u) = \cos u \frac{du}{dx}, \quad \frac{d}{dx}(\cos u) = -\sin u \frac{du}{dx}, \quad \frac{d}{dx}(\tan u) = \sec^2 u \frac{du}{dx}$ $\frac{d}{dx}(e^u) = e^u \frac{du}{dx}, \quad \frac{d}{dx}(\ln u) = \frac{1}{u}\frac{du}{dx}, \quad \frac{d}{dx}(\arctan u) = \frac{1}{1 + u^2}\frac{du}{dx}$

Critical Points & Extrema Classification

A critical point of a differentiable function $y = f(x)$ occurs where $f'(x) = 0$ or where $f'(x)$ is undefined:

• First Derivative Test:
  • $f'(x)$ changes from positive to negative at $x = c \implies$ Local Maximum
  • $f'(x)$ changes from negative to positive at $x = c \implies$ Local Minimum
  • $f'(x)$ maintains the same sign $\implies$ Horizontal Inflection Point
• Second Derivative Test:
  • If $f'(c) = 0$ and $f''(c) < 0 \implies$ Local Maximum (curve is concave downward)
  • If $f'(c) = 0$ and $f''(c) > 0 \implies$ Local Minimum (curve is concave upward)
  • If $f'(c) = 0$ and $f''(c) = 0 \implies$ Test is inconclusive; use First Derivative Test.
• Point of Inflection: A point where the concavity changes sign. A necessary condition for an inflection point is $f''(x) = 0$ or undefined, with a proven sign change of $f''(x)$ across the point.

---

Related Rates in Civil Engineering

Related rates problems model physical systems where multiple interconnected geometric parameters change dynamically over time $t$. The time derivatives are linked via implicit differentiation using the chain rule.

Systematic 4-Step Solution Method

1. Geometric Sketch & Parameter Identification: Assign variable symbols to all dynamic quantities and identify constants.
2. Formulate Constraint Equation: Write a single geometric, trigonometric, or volumetric equation relating the variables (e.g., Pythagorean theorem, similar triangles, volume formula).
3. Implicit Differentiation: Differentiate the entire equation with respect to time $t$ ($\frac{d}{dt}$).
4. Substitution of Instantaneous Values: Substitute known numerical values and given rates only after differentiation to solve for the target rate.

Standard CELE Related Rates Scenarios

1. Conical Tank Draining: Water draining from an inverted right circular cone of fixed height $H$ and base radius $R$:
   • By similar triangles: $\frac{r}{h} = \frac{R}{H} \implies r = \left(\frac{R}{H}\right)h$
   • Volume: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \left(\frac{R}{H}\right)^2 h^3$
   • Differentiation: $\frac{dV}{dt} = \pi \left(\frac{R}{H}\right)^2 h^2 \frac{dh}{dt}$
2. Sliding Structural Ladder: A ladder of constant length $L$ sliding down a vertical wall:
   • Constraint: $x^2 + y^2 = L^2$
   • Differentiation: $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0 \implies \frac{dy}{dt} = -\frac{x}{y}\frac{dx}{dt}$

---

Integral Calculus & Integration Techniques

Core Integration Strategies

• Integration by Substitution ($u$-sub): Transforms $\int f(g(x))g'(x)\,dx$ into $\int f(u)\,du$.
• Integration by Parts: $\int u\,dv = uv - \int v\,du$ (Follow the LIATE mnemonic to assign $u$: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).
• Trigonometric Substitutions:
  • For $\sqrt{a^2 - x^2}$: Substitute $x = a\sin\theta \implies dx = a\cos\theta\,d\theta$
  • For $\sqrt{a^2 + x^2}$: Substitute $x = a\tan\theta \implies dx = a\sec^2\theta\,d\theta$
  • For $\sqrt{x^2 - a^2}$: Substitute $x = a\sec\theta \implies dx = a\sec\theta\tan\theta\,d\theta$
• Partial Fractions Decomposition: Resolves rational functions $P(x)/Q(x)$ into sums of simpler fractions with linear or irreducible quadratic denominators.

---

Definite Integrals: Areas, Volumes of Revolution & Centroids

1. Area Between Plane Curves

For regions bounded by upper curve $y_2 = f(x)$ and lower curve $y_1 = g(x)$ from $x = a$ to $x = b$: $A = \int_a^b [f(x) - g(x)]\,dx$ Using horizontal strips bounded by right curve $x_2 = f(y)$ and left curve $x_1 = g(y)$ from $y = c$ to $y = d$: $A = \int_c^d [f(y) - g(y)]\,dy$

---

2. Volumes of Solids of Revolution

---

3. Centroids of Plane Areas

The coordinates of the centroid $(\bar{x}, \bar{y})$ of a planar lamina with area $A$ are defined by first moments of area: $\bar{x} = \frac{Q_y}{A} = \frac{1}{A} \int \tilde{x}\,dA, \quad \bar{y} = \frac{Q_x}{A} = \frac{1}{A} \int \tilde{y}\,dA$

• Using vertical strips of width $dx$: $A = \int_a^b [y_2 - y_1]\,dx, \quad \bar{x} = \frac{1}{A} \int_a^b x[y_2 - y_1]\,dx, \quad \bar{y} = \frac{1}{2A} \int_a^b [y_2^2 - y_1^2]\,dx$

4. Theorems of Pappus-Guldinus

Pappus's theorems provide extremely fast shortcuts for volumes and surface areas of revolution:

• First Theorem (Surface Area): The surface area generated by rotating a plane curve of length $L$ about an external coplanar axis is equal to the curve length multiplied by the distance traveled by its centroid: $S = 2\pi \bar{r} L$
• Second Theorem (Volume): The volume of a solid of revolution generated by rotating a plane area $A$ about an external coplanar axis is equal to the area multiplied by the distance traveled by its area centroid: $V = 2\pi \bar{r} A$

---

Step-by-Step Worked Problems

Worked Example: Related Rates in Water Storage Tank

Problem: A municipal elevated conical drainage hopper has a top diameter of $6.0\text{ m}$ (radius $R = 3.0\text{ m}$) and a total height $H = 9.0\text{ m}$. Water is pumped into the hopper at a constant rate of $1.80\text{ m}^3/\text{min}$, while simultaneously draining out of the bottom orifice at a rate of $0.60\text{ m}^3/\text{min}$. At what exact rate is the water surface rising when the instantaneous water depth in the hopper is $4.5\text{ m}$?

Solution:

1. Net rate of volumetric accumulation: $\frac{dV}{dt} = Q_{\text{in}} - Q_{\text{out}} = 1.80 - 0.60 = +1.20\text{ m}^3/\text{min}$
2. Geometric relationship via similar triangles: $\frac{r}{h} = \frac{R}{H} = \frac{3.0}{9.0} = \frac{1}{3} \implies r = \frac{h}{3}$
3. Express volume solely as a function of depth $h$: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \left(\frac{h}{3}\right)^2 h = \frac{\pi}{27} h^3$
4. Differentiate implicitly with respect to time $t$: $\frac{dV}{dt} = \frac{\pi}{27} \cdot 3h^2 \frac{dh}{dt} = \frac{\pi h^2}{9} \frac{dh}{dt}$
5. Substitute known parameters ($h = 4.5\text{ m}$ and $\frac{dV}{dt} = 1.20\text{ m}^3/\text{min}$): $1.20 = \frac{\pi (4.5)^2}{9} \frac{dh}{dt} = \frac{\pi (20.25)}{9} \frac{dh}{dt} = 2.25\pi \frac{dh}{dt}$ $\frac{dh}{dt} = \frac{1.20}{2.25\pi} = \frac{1.20}{7.06858} \approx 0.1698\text{ m/min} \approx 0.170\text{ m/min}$
6. The water surface is rising at a rate of $0.170\text{ m/min}$ ($17.0\text{ cm/min}$).

---

CELE Board Exam Traps & Strategic Checklists