7.4 Area Under Curves & First-Order Differential Equations

Area Under Curves & First-Order Differential Equations

The practical power of integral calculus is demonstrated across two vital engineering domains: determining planar areas bounded by geometric curves and solving differential equations that govern dynamic physical systems. Both topics constitute core components of intermediate mathematics (FSc Part II) and feature prominently on the UET ECAT. Engineering entrance questions test not only your ability to perform rigorous integration, but also your proficiency with geometric shortcuts (such as Archimedes' parabolic area formula) and rapid differential equation classification. This section explores geometric area calculations, establishes rigorous sign-management principles, defines differential order and degree, and provides systematic solution blueprints for first-order ordinary differential equations.

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1. Geometric Interpretation of Definite Integrals & Planar Area

Net Signed Area vs. Geometric Physical Area

When a curve $y = f(x)$ lies entirely above the $x$-axis on $[a, b]$, the definite integral $\int_a^b f(x) dx$ evaluates the exact geometric area enclosed between the curve, the $x$-axis, and the vertical lines $x = a$ and $x = b$. However, if $f(x)$ drops below the $x$-axis, the function values are negative, and the Riemann sum evaluates these regions with a negative algebraic sign. The definite integral calculates the net signed area:

$\text{Net Signed Area} = \int_a^b f(x) dx = A_{\text{above}} - A_{\text{below}}$

To compute the true total geometric area $A$, one must integrate the absolute value of the function:

$A = \int_a^b |f(x)| dx$

Systematic Three-Step Procedure for Single-Curve Area

1. Find Real Roots: Solve $f(x) = 0$ on the interval $[a, b]$ to identify all $x$-intercepts where the curve crosses the axis.
2. Partition Interval: Subdivide $[a, b]$ at each root into subintervals where $f(x)$ maintains a constant sign.
3. Sum Absolute Magnitudes: Integrate $f(x)$ over each subinterval, taking the absolute value of each resulting quantity: $A = \left| \int_a^{c_1} f(x) dx \right| + \left| \int_{c_1}^{c_2} f(x) dx \right| + \dots + \left| \int_{c_k}^b f(x) dx \right|$

ECAT Walkthrough Example: Find the total area bounded by $y = \sin x$ and the $x$-axis from $x = 0$ to $x = 2\pi$.

• If you blindly integrate without splitting: $\int_0^{2\pi} \sin x dx = [-\cos x]_0^{2\pi} = -1 - (-1) = 0$, which is geometrically absurd!
• The root on $(0, 2\pi)$ is $x = \pi$.
• Partition at $\pi$: $A_1 = \int_0^\pi \sin x dx = [-\cos x]_0^\pi = -(-1) - (-1) = 2$ $A_2 = \int_\pi^{2\pi} \sin x dx = [-\cos x]_\pi^{2\pi} = -1 - (-(-1)) = -2 \implies |A_2| = 2$
• Total geometric area: $A = 2 + 2 = 4$ square units.

Area Bounded with Respect to the $y$-Axis

When a curve is given by $x = g(y)$, bounded by the $y$-axis and horizontal lines $y = c$ and $y = d$:

$A = \int_c^d |g(y)| dy$

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2. Area Bounded Between Two Curves

General Formulation

If $f(x)$ and $g(x)$ are continuous functions such that $f(x) \ge g(x)$ for all $x \in [a, b]$, the area bounded between the two curves from $x = a$ to $x = b$ is the integral of the vertical difference:

$A = \int_a^b [y_{\text{upper}} - y_{\text{lower}}] dx = \int_a^b [f(x) - g(x)] dx$

Critical Insight: This formula holds strictly true regardless of whether both curves lie above the $x$-axis, both lie below the $x$-axis, or one lies above and one below. The vertical height of a representative strip is always $y_{\text{upper}} - y_{\text{lower}}$.

Integration Along the $y$-Axis

When curves are expressed more simply as functions of $y$, such as horizontal parabolas $x = f(y)$, vertical strips are replaced with horizontal strips of width $dy$ and length $x_{\text{right}} - x_{\text{left}}$:

$A = \int_c^d [x_{\text{right}} - x_{\text{left}}] dy$

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3. High-Speed ECAT Area Shortcuts (Archimedes' Theorems)

Manually solving for intersection limits and integrating quadratic polynomials consumes 2 to 3 minutes. The following closed-form geometric shortcuts allow candidates to solve standard ECAT parabola area questions in under 15 seconds.

1. Parabola Bounded by a Chord Perpendicular to its Axis (Archimedes' Formula)

The area of a parabolic segment bounded by the parabola and a chord perpendicular to its axis of symmetry equals two-thirds of the area of the circumscribed bounding rectangle:

$A = \frac{2}{3} \times \text{base} \times \text{height}$

Application: Area bounded by $y^2 = 4ax$ and its Latus Rectum ($x = a$):

• Base (length of latus rectum) $= 4a$.
• Height (distance from vertex $(0,0)$ to chord $x = a$) $= a$.
• Area $A = \frac{2}{3}(4a)(a) = \frac{8}{3}a^2$.

2. Area Bounded by Parabola $y^2 = 4ax$ and Line $y = mx$

$A = \frac{8 a^2}{3 m^3}$

Example: Find the area bounded by $y^2 = 8x$ and $y = 2x$.

• Here $4a = 8 \implies a = 2$, and $m = 2$.
• $A = \frac{8(2^2)}{3(2^3)} = \frac{8(4)}{3(8)} = \frac{4}{3}$ square units. Solved in 5 seconds!

3. Area Bounded by Two Orthogonal Parabolas $y^2 = 4ax$ and $x^2 = 4by$

$A = \frac{16 a b}{3}$

Example: Find the area enclosed between $y^2 = 4x$ and $x^2 = 4y$.

• Here $4a = 4 \implies a = 1$ and $4b = 4 \implies b = 1$.
• $A = \frac{16(1)(1)}{3} = \frac{16}{3}$ square units.

4. Area Bounded by Parabola $y = ax^2$ and Line $y = c$

$A = \frac{4}{3} c \sqrt{\frac{c}{a}}$

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4. Introduction to Differential Equations: Order & Degree

A differential equation (DE) is an algebraic equation that establishes a relationship between an independent variable $x$, a dependent variable $y$, and one or more derivatives of $y$ with respect to $x$:

$F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \dots, \frac{d^n y}{dx^n}\right) = 0$

Definitions of Order and Degree

1. Order: The order of a differential equation is the order of the highest derivative appearing anywhere in the equation.
2. Degree: The degree of a differential equation is the power (positive integer exponent) of the highest-order derivative, after the differential equation has been converted into a rational integral form (meaning all fractional powers and radical signs on derivatives have been cleared).

ECAT Trap Alert: Degree is defined only when the equation can be written as a polynomial in its derivatives. If derivatives appear inside transcendental functions (e.g., $\sin\left(\frac{dy}{dx}\right) + y = 0$ or $e^{y'} = x$), the degree is undefined, though the order remains well-defined!

Walkthrough: Determining Order and Degree

Determine the order and degree of: $\left[ 1 + \left(\frac{dy}{dx}\right)^2 \right]^{3/2} = 5 \frac{d^2y}{dx^2}$

1. Identify the highest derivative: $\frac{d^2y}{dx^2}$, which has order $2$.
2. Clear the fractional exponent $\frac{3}{2}$ by squaring both sides: $\left[ 1 + \left(\frac{dy}{dx}\right)^2 \right]^3 = 25 \left(\frac{d^2y}{dx^2}\right)^2$
3. The highest-order derivative $\frac{d^2y}{dx^2}$ now carries power $2$.
4. Order = 2, Degree = 2.

General Solution vs. Particular Solution

• General Solution: A solution containing $n$ independent arbitrary constants, where $n$ is equal to the order of the differential equation. A first-order DE contains exactly $1$ arbitrary constant $c$.
• Particular Solution: A solution obtained from the general solution by assigning specific numerical values to the arbitrary constants using given initial or boundary conditions (e.g., $y(0) = 1$).

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5. First-Order Differential Equations: Three Primary Forms

Most first-order differential equations in the Intermediate syllabus fall into one of three standard categories:

Category 1: Variable Separable Differential Equations

An equation is variable separable if it can be algebraically factored such that all terms involving $y$ multiply $dy$ on one side, and all terms involving $x$ multiply $dx$ on the opposite side:

$\frac{dy}{dx} = g(x) h(y) \implies \frac{1}{h(y)} dy = g(x) dx$

Integrate both sides directly: $\int \frac{1}{h(y)} dy = \int g(x) dx + c$

Example: Solve $\frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}$.

• Separate: $\frac{dy}{1 + y^2} = \frac{dx}{1 + x^2}$.
• Integrate: $\arctan y = \arctan x + c$.
• Using the tangent addition formula, this simplifies to $y = \frac{x + C}{1 - Cx}$.

Category 2: Homogeneous First-Order Differential Equations

A first-order equation $\frac{dy}{dx} = \frac{f(x, y)}{g(x, y)}$ is homogeneous if both $f(x, y)$ and $g(x, y)$ are homogeneous polynomials of the same degree $n$, meaning $f(\lambda x, \lambda y) = \lambda^n f(x, y)$. Alternatively, $\frac{dy}{dx} = F\left(\frac{y}{x}\right)$.

Standard Substitution: Let $y = vx$. Differentiating with respect to $x$ via the product rule: $\frac{dy}{dx} = v + x \frac{dv}{dx}$

Substitute into the differential equation: $v + x \frac{dv}{dx} = F(v) \implies x \frac{dv}{dx} = F(v) - v$

This transforms the homogeneous equation into a variable-separable equation in $v$ and $x$: $\frac{dv}{F(v) - v} = \frac{dx}{x}$ After integrating, replace $v$ with $\frac{y}{x}$ to express the solution in terms of the original variables.

Category 3: Linear First-Order Differential Equations (Leibniz Form)

A differential equation is linear of first order if it can be written in the canonical form:

$\frac{dy}{dx} + P(x) y = Q(x)$

where $P(x)$ and $Q(x)$ are continuous functions of $x$ alone (or constants), and $y$ appears strictly to the first power with no products involving $y$ and $\frac{dy}{dx}$.

The Integrating Factor (I.F.) Method

1. Identify $P(x)$ and $Q(x)$ from standard form.
2. Compute the Integrating Factor $I(x)$: $I(x) = e^{\int P(x) dx}$
3. Write the General Solution: $y \cdot I(x) = \int Q(x) I(x) dx + c$

Proof: Multiply the canonical equation by $I(x) = e^{\int P(x) dx}$: $e^{\int P(x) dx} \frac{dy}{dx} + P(x) e^{\int P(x) dx} y = Q(x) e^{\int P(x) dx}$ By the product rule, the left-hand side is the exact derivative $\frac{d}{dx}\left[ y e^{\int P(x) dx} \right]$. Integrating both sides with respect to $x$ yields $y \cdot I(x) = \int Q(x) I(x) dx + c$.

Linear Form with Dependent Variable $x$

If $\frac{dy}{dx}$ is non-linear in $y$ but linear in $x$, invert to find $\frac{dx}{dy}$: $\frac{dx}{dy} + P(y) x = Q(y) \implies I(y) = e^{\int P(y) dy}, \quad x \cdot I(y) = \int Q(y) I(y) dy + c$

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6. Differential Equation Diagnostic & Classification Table