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MAT 266: Calculus for Engineers II - Study Guide

MAT 266: Calculus for Engineers II - Study Guide






MAT 266: Calculus for Engineers II

A comprehensive outline of key concepts, formulas, and applications.





I. Methods of Integration


Substitution Method (u-Substitution)

Used to reverse the chain rule. Identify an "inner" function $g(x)$ whose derivative $g'(x)$ is also present.


Let $u = g(x)$, then $du = g'(x)dx$.
$$ \int f(g(x))g'(x)dx = \int f(u)du $$



Integration by Parts

Used to reverse the product rule. A common mnemonic to choose $u$ is LIATE (Logarithmic, Inverse Trig, Algebraic, Trigonometric, Exponential).


$$ \int u \, dv = uv - \int v \, du $$



Trigonometric Substitution

Used for integrals containing specific forms of square roots.


For $\sqrt{a^2 - x^2}$, let $x = a\sin\theta$, then $dx = a\cos\theta d\theta$. Identity: $1 - \sin^2\theta = \cos^2\theta$.

For $\sqrt{a^2 + x^2}$, let $x = a\tan\theta$, then $dx = a\sec^2\theta d\theta$. Identity: $1 + \tan^2\theta = \sec^2\theta$.

For $\sqrt{x^2 - a^2}$, let $x = a\sec\theta$, then $dx = a\sec\theta\tan\theta d\theta$. Identity: $\sec^2\theta - 1 = \tan^2\theta$.



Integration by Partial Fractions

Used to integrate rational functions $\frac{P(x)}{Q(x)}$ by decomposing them into simpler, integrable fractions based on the factors of the denominator $Q(x)$.




II. Numerical & Improper Integration


Numerical Integration (Approximation)

For a definite integral $\int_a^b f(x)dx$ with $\Delta x = \frac{b-a}{n}$.


Midpoint Rule: $$ M_n = \Delta x [f(\bar{x}_1) + f(\bar{x}_2) + \dots + f(\bar{x}_n)] $$

Trapezoidal Rule: $$ T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + \dots + 2f(x_{n-1}) + f(x_n)] $$

Simpson's Rule (n must be even): $$ S_n = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)] $$



Improper Integrals

Integrals over an infinite interval or where the function has an infinite discontinuity.


Type 1 (Infinite Interval): $$ \int_a^\infty f(x)dx = \lim_{t \to \infty} \int_a^t f(x)dx $$

Type 2 (Discontinuity at b): $$ \int_a^b f(x)dx = \lim_{t \to b^-} \int_a^t f(x)dx $$





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