MA200 Integral Calculus: Area Between Curves, Volumes of Revolution, and Arc Length

Foundations: Area Between Curves and Riemann Sum Generalization

• Riemann Sum Framework: Definite integrals evaluate net signed area. Pure geometric area between curves is always non-negative.

• Area Between Curves: For continuous functions with $f(x) \ge g(x)$ on $[a, b]$:   $A = \int_a^b (f(x) - g(x))\,dx$

• Intersecting Regions: Partition the integral at intersection points to maintain $(\text{upper curve}) - (\text{lower curve})$.

  • Example: Enclosing $y = x^3$ and $y = 4x$ over $[-1, 1]$ splits at $x = 0$:     $A = \int_{-1}^0 (x^3 - 4x)\,dx + \int_0^1 (4x - x^3)\,dx = \frac{7}{4} + \frac{7}{4} = \frac{7}{2}$

• Standard Evaluations:

  • Enclosed by $g(x) = \sqrt{x}$ and $f(x) = x$ over $[0, 1]$: $A = \int_0^1 (\sqrt{x} - x)\,dx = \frac{1}{6}$

  • Enclosed by $f(x) = 4^x$, $g(x) = 2^x$, and $x = 1$: $A = \int_0^1 (4^x - 2^x)\,dx = \frac{1}{2\ln(2)}$

Volume by Slicing and Cross-Sectional Area

• Slicing Principle: Integrating parallel cross-sectional slices of area $A(x)$ along the $x$-axis yields total volume:   $V = \int_a^b A(x)\,dx$

• Uniform Cross Section: For a constant $2 \times 2$ square cross section on $[0, 10]$, $V = \int_0^{10} 4\,dx = 40\,\text{units}^3$.

• Square Pyramid: Apex at origin, height $h = 10$, base side $b = 6$. Side length scales as $s(x) = \frac{3}{5}x$:   $A(x) = \frac{9}{25}x^2 \implies V = \int_0^{10} \frac{9}{25}x^2\,dx = 120\,\text{units}^3$

• Right Circular Cone: Apex at origin, height $h = 10$, base radius $r = 5$. Radius scales as $r(x) = \frac{1}{2}x$:   $A(x) = \frac{\pi x^2}{4} \implies V = \int_0^{10} \frac{\pi x^2}{4}\,dx = \frac{250\pi}{3}\,\text{units}^3$

Solids of Revolution and the Disk Method

• Disk Method Formula: Rotating $y = f(x)$ around the $x$-axis forms cylindrical disks of radius $r = f(x)$:   $V = \int_a^b \pi (f(x))^2\,dx$

• Applications:

  • Cone verification: Revolving $y = \frac{1}{2}x$ on $[0, 10]$ gives $V = \pi \int_0^{10} \frac{x^2}{4}\,dx = \frac{250\pi}{3}\,\text{units}^3$.

  • Rational curve $y = \frac{1}{x}$ on $[1, 2]$: $V = \pi \int_1^2 x^{-2}\,dx = \frac{\pi}{2}\,\text{units}^3$.

  • Radical curve $y = \sqrt{x}$ on $[0, 4]$: $V = \pi \int_0^4 x\,dx = 8\pi\,\text{units}^3$.

Differentiation Rules and Geometric Foundations

• Calculus Rules:

  • Product Rule: $\frac{d}{dx}(uv) = u'v + vu'$

  • Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - v'u}{v^2}$

  • Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$

  • Exponentials: $\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(a^x) = \ln(a) \cdot a^x$, and $\frac{d}{dx}(2^{-x}) = -\ln(2) \cdot 2^{-x}$

• Distance Relations:

  • Pythagorean Theorem: $c = \sqrt{a^2 + b^2}$

  • Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Arc Length: Derivation and Rigorous Formulation

• Approximation: Segment sum of secant chords: $\sum_{i=1}^n \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2}$.

• Definite Integral Derivation: Factoring $\Delta x_i$ forms a Riemann sum converging to:   $L = \lim_{n \rightarrow \infty} \sum_{i=1}^n \sqrt{1 + \left(\frac{\Delta y_i}{\Delta x_i}\right)^2} \Delta x_i = \int_a^b \sqrt{1 + (f'(x))^2}\,dx$

• Condition: $f'(x)$ must be continuous on $[a, b]$.

Arc Length Computations and Analytical Techniques

• Parabola $y = \frac{1}{2}x^2$ over $[0, 3]$:   $L = \int_0^3 \sqrt{1 + x^2}\,dx \approx 5.65$

• Linear function $y = 2x + 1$ over $[0, 2]$:   $L = \int_0^2 \sqrt{1 + 2^2}\,dx = 2\sqrt{5}$

• Power function $y = x^{3/2}$ over $[0, 4]$ (via $u = 1 + \frac{9}{4}x$):   $L = \int_0^4 \sqrt{1 + \frac{9}{4}x}\,dx = \frac{8}{27}(10^{3/2} - 1) \approx 9.07$

• Logarithmic function $f(x) = \frac{x^2}{8} - \ln(x)$ over $[1, 2]$:

  • Radicand forms a perfect square: $1 + (f'(x))^2 = \left(\frac{x}{4} + \frac{1}{x}\right)^2$

  • Integral evaluation: $$L = \int_1^2 \left(\frac{x}{4} + \frac{1}{x}\right)\,dx = \frac{3}{8} + \ln(2) \approx