If a region is bounded between two functions \( f(x) \) and \( g(x) \) on the interval \([a, b]\), where \( f(x) \geq g(x) \), the area is:
\[ A = \int_a^b \left[ f(x) - g(x) \right] \, dx \]
If a region is bounded between two functions \( f(y) \) and \( g(y) \) on the interval \([c, d]\), where \( f(y) \geq g(y) \), the area is:
\[ A = \int_c^d \left[ f(y) - g(y) \right] \, dy \]
Trapezoidal Rule:
\[ \int_a^b f(x) \, dx \approx \frac{b - a}{2n} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] \]
Simpson’s Rule:
\[ \int_a^b f(x) \, dx \approx \frac{b - a}{3n} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n) \right] \]
Disk Method (about the x-axis):
\[ V = \pi \int_a^b \left[ f(x) \right]^2 \, dx \]
Disk Method (about the y-axis):
\[ V = \pi \int_c^d \left[ g(y) \right]^2 \, dy \]
Washer Method:
\[ V = \pi \int_a^b \left( R(x)^2 - r(x)^2 \right) \, dx \]
Shell Method (about the y-axis):
\[ V = 2\pi \int_a^b x \cdot f(x) \, dx \]
Shell Method (about the x-axis):
\[ V = 2\pi \int_c^d y \cdot g(y) \, dy \]
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \] \[ \int \frac{1}{x} \, dx = \ln|x| + C \] \[ \int e^x \, dx = e^x + C \] \[ \int a^x \, dx = \frac{a^x}{\ln a} + C \]
\[ \int u \, dv = uv - \int v \, du \]
Common identities used:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2}, \quad \cos^2 x = \frac{1 + \cos(2x)}{2} \]
Use substitutions to simplify integrals of the following forms:
\[ \sqrt{a^2 - x^2} \Rightarrow x = a \sin \theta \] \[ \sqrt{a^2 + x^2} \Rightarrow x = a \tan \theta \] \[ \sqrt{x^2 - a^2} \Rightarrow x = a \sec \theta \]
For rational functions where the degree of the numerator is less than the degree of the denominator:
\[ \frac{P(x)}{(x - r_1)(x - r_2)} = \frac{A}{x - r_1} + \frac{B}{x - r_2} \]
Used to evaluate indeterminate forms:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \quad \text{if } \frac{0}{0} \text{ or } \frac{\infty}{\infty} \]
Type I: Infinite interval:
\[ \int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx \]
Type II: Discontinuous integrand:
\[ \int_a^b f(x) \, dx = \lim_{t \to c^-} \int_a^t f(x) \, dx + \lim_{t \to c^+} \int_t^b f(x) \, dx \]
h3>SequencesA sequence is a list of terms \( \{a_n\} = a_1, a_2, a_3, \ldots \) indexed by natural numbers. It converges if \( \lim_{n \to \infty} a_n = L \) for some finite value \( L \); otherwise, it diverges.
An infinite series is the sum \( \sum_{n=1}^{\infty} a_n \). It converges if the sequence of partial sums \( S_n = \sum_{k=1}^{n} a_k \) has a finite limit.
A geometric series is \( \sum_{n=0}^{\infty} ar^n \). It converges if \( |r| < 1 \), with sum \( \frac{a}{1 - r} \); otherwise, it diverges.
If \( \lim_{n \to \infty} a_n \neq 0 \), then \( \sum a_n \) diverges. If the limit is 0, the test is inconclusive.
If \( f(n) = a_n \) is positive, continuous, and decreasing for \( n \geq N \), then \( \sum_{n=N}^{\infty} a_n \) converges if and only if \( \int_N^{\infty} f(x) \, dx \) converges.
A p-series is \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). It converges if \( p > 1 \); diverges if \( p \leq 1 \).
For positive terms: If \( 0 \leq a_n \leq b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) converges. If \( a_n \geq b_n \geq 0 \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges.
If \( a_n > 0 \), \( b_n > 0 \), and \[ \lim_{n \to \infty} \frac{a_n}{b_n} = c, \quad 0 < c < \infty, \] then \( \sum a_n \) and \( \sum b_n \) both converge or both diverge.
An alternating series \( \sum (-1)^{n+1} b_n \) converges if: (1) \( b_n \) is decreasing, and (2) \( \lim_{n \to \infty} b_n = 0 \).
Used to test convergence of an infinite series \( \sum a_n \). Let: \[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
Then:
Used to test convergence of an infinite series \( \sum a_n \). Let: \[ L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \]
Then:
A power series has the form \( \sum_{n=0}^{\infty} a_n(x - c)^n \). It converges when \( |x - c| < R \), where \( R \) is the radius of convergence. It diverges when \( |x - c| > R \). Behavior at endpoints must be tested separately.
The Taylor series of \( f(x) \) centered at \( a \) is: \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n \] The Maclaurin series is the Taylor series centered at \( a = 0 \).
The remainder after \( n \) terms is given by Taylor’s Theorem: \[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1} \] for some \( c \) between \( a \) and \( x \). This gives an upper bound on the error when approximating \( f(x) \) by its \( n \)-th degree Taylor polynomial.
\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}, \quad \text{all } x \] \[ \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}, \quad \text{all } x \] \[ \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}, \quad \text{all } x \] \[ \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n, \quad |x| < 1 \] \[ \ln(1 + x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}, \quad |x| < 1 \]
A first-order differential equation involves the first derivative of an unknown function.
General form: \[ \frac{dy}{dx} = f(x, y) \]
If the equation is separable, rewrite it in the form: \[ \frac{1}{h(y)}\, dy = g(x)\, dx \] Then integrate both sides: \[ \int \frac{1}{h(y)}\, dy = \int g(x)\, dx \] Solve for \( y \) if possible.
If the equation is already separated: \[ \frac{dy}{dx} = f(x) \] Integrate directly: \[ y = \int f(x)\, dx + C \]
Standard form (vertical): \( (x - h)^2 = 4p(y - k) \)
Standard form (horizontal): \( (y - k)^2 = 4p(x - h) \)
Standard form: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] Foci distance: \( c = \sqrt{a^2 - b^2} \) Eccentricity: \( e = \frac{c}{a} \)
Standard forms: \[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \quad \text{or} \quad \frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1 \] Foci distance: \( c = \sqrt{a^2 + b^2} \) Eccentricity: \( e = \frac{c}{a} \)
Defined by: \[ x = f(t), \quad y = g(t) \]
Slope: \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Second derivative: \[ \frac{d^2y}{dx^2} = \frac{d}{dt} \left( \frac{dy}{dx} \right) \div \frac{dx}{dt} \]
Arc length: \[ L = \int_a^b \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 }\, dt \]
Surface area (rotation about x-axis): \[ S = \int_a^b 2\pi y(t) \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 }\, dt \]
Area under curve: \[ A = \int_a^b y(t) \cdot \frac{dx}{dt} \, dt \]
Conversion: \[ x = r \cos \theta, \quad y = r \sin \theta, \quad r^2 = x^2 + y^2, \quad \tan \theta = \frac{y}{x} \]
Area enclosed: \[ A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 \, d\theta \]
Arc length: \[ L = \int_{\alpha}^{\beta} \sqrt{[r(\theta)]^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \]
Surface area (rotation about initial line): \[ S = \int_{\alpha}^{\beta} 2\pi r(\theta) \sin \theta \cdot \sqrt{[r(\theta)]^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \]