Unit 6 Formula Sheet

Integration and Accumulation of Change — key formulas to know

← All learning targets

Riemann Sums & the Limit Definition

Step size:$$\Delta x = \dfrac{b-a}{n}$$
Right endpoint:$$x_i^* = a + i\,\Delta x$$
Left endpoint:$$x_i^* = a + (i-1)\,\Delta x$$
Midpoint:$$x_i^* = a + (i-\tfrac12)\,\Delta x$$
Riemann sum:$$\sum_{i=1}^{n} f(x_i^*)\,\Delta x$$
Definite integral as a limit:$$\int_{a}^{b} f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\,\Delta x$$

Summation Formulas

Sum of first n integers:$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
Sum of squares:$$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$
Sum of cubes:$$\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^{2}$$
Sum of a constant:$$\sum_{i=1}^{n} c = nc$$
Constant multiple:$$\sum_{i=1}^{n} c\,a_i = c\sum_{i=1}^{n} a_i$$
Sum / difference:$$\sum_{i=1}^{n} (a_i \pm b_i) = \sum_{i=1}^{n} a_i \pm \sum_{i=1}^{n} b_i$$

Trapezoidal Rule

n subintervals:$$\int_{a}^{b} f(x)\,dx \approx \frac{\Delta x}{2}\Big[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\Big]$$

Properties of Definite Integrals

Zero-width interval:$$\int_{a}^{a} f(x)\,dx = 0$$
Reversing limits:$$\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx$$
Additivity:$$\int_{a}^{b} f(x)\,dx + \int_{b}^{c} f(x)\,dx = \int_{a}^{c} f(x)\,dx$$
Constant multiple:$$\int_{a}^{b} c\,f(x)\,dx = c\int_{a}^{b} f(x)\,dx$$
Sum / difference:$$\int_{a}^{b} \big[f(x) \pm g(x)\big]\,dx = \int_{a}^{b} f(x)\,dx \pm \int_{a}^{b} g(x)\,dx$$
Even function:$$\int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx \quad (f\text{ even})$$
Odd function:$$\int_{-a}^{a} f(x)\,dx = 0 \quad (f\text{ odd})$$

Fundamental Theorem of Calculus

FTC Part 1 (evaluate):$$\int_{a}^{b} f(x)\,dx = F(b) - F(a), \quad F' = f$$
FTC Part 2 (differentiate):$$\frac{d}{dx}\int_{a}^{x} f(t)\,dt = f(x)$$
FTC + chain rule:$$\frac{d}{dx}\int_{a}^{g(x)} f(t)\,dt = f(g(x))\,g'(x)$$
Two variable limits:$$\frac{d}{dx}\int_{h(x)}^{g(x)} f(t)\,dt = f(g(x))g'(x) - f(h(x))h'(x)$$

Basic Antiderivatives (Power, Log, Exponential)

Power rule:$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
Reciprocal:$$\int \frac{1}{x}\,dx = \ln|x| + C$$
Exponential:$$\int e^x\,dx = e^x + C$$
General exponential:$$\int a^x\,dx = \frac{a^x}{\ln a} + C$$

Basic Antiderivatives (Trig & Inverse Trig)

Sine:$$\int \sin x\,dx = -\cos x + C$$
Cosine:$$\int \cos x\,dx = \sin x + C$$
Secant squared:$$\int \sec^2 x\,dx = \tan x + C$$
Arctangent:$$\int \frac{1}{1+x^2}\,dx = \tan^{-1}x + C$$
Arcsine:$$\int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x + C$$

U-Substitution

Substitution:$$\int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x)$$
Differential:$$du = g'(x)\,dx$$
Definite integral:$$\int_{a}^{b} f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$$

Integration by Parts

Parts formula:$$\int u\,dv = uv - \int v\,du$$
Choosing u (LIATE):$$\text{Log} \to \text{Inverse trig} \to \text{Algebraic} \to \text{Trig} \to \text{Exponential}$$

Partial Fractions & Long Division

Distinct linear factors:$$\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$
Long division first:$$\text{if } \deg(\text{numerator}) \ge \deg(\text{denominator})$$

Improper Integrals

Infinite upper limit:$$\int_{a}^{\infty} f(x)\,dx = \lim_{b\to\infty}\int_{a}^{b} f(x)\,dx$$
Infinite lower limit:$$\int_{-\infty}^{b} f(x)\,dx = \lim_{a\to-\infty}\int_{a}^{b} f(x)\,dx$$
p-test:$$\int_{1}^{\infty} \frac{1}{x^p}\,dx \text{ converges } \iff p > 1$$

Applications

Average value:$$f_{\text{avg}} = \frac{1}{b-a}\int_{a}^{b} f(x)\,dx$$
Displacement:$$\int_{a}^{b} v(t)\,dt = s(b) - s(a)$$
Total distance:$$\int_{a}^{b} \big|v(t)\big|\,dt$$
Area between curves:$$\int_{a}^{b} \big[\text{top} - \text{bottom}\big]\,dx$$
Disk volume:$$V = \pi\int_{a}^{b} \big[f(x)\big]^2\,dx$$
Washer volume:$$V = \pi\int_{a}^{b} \big[R(x)^2 - r(x)^2\big]\,dx$$
Arc length:$$L = \int_{a}^{b} \sqrt{1 + \big[f'(x)\big]^2}\,dx$$