Calculus 2 Final Exam Cheat Sheet

University of Arkansas - MATH 25004 (Spring 2025)

1. Integration Techniques

Substitution

• Formula: ( \int f(g(x))g'(x),dx \Rightarrow u = g(x),\ du = g'(x)dx )
• Example: ( \int e^{3x+2}dx ), let ( u = 3x + 2 )

Integration by Parts

• Formula: ( \int u,dv = uv - \int v,du )
• LIATE rule for choosing ( u ) : Log, Inverse trig, Algebraic, Trig, Exponential
• Example: ( \int \sqrt{x}\ln(x)dx )

Trig Substitution

• ( \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 )
• Example: ( \int x^3\sqrt{16 - x^2},dx )

Trig Integrals

• Identities: ( \sin^2x = 1 - \cos^2x, \tan^2x + 1 = \sec^2x )
• Example: ( \int \tan^4(\theta)\sec^6(\theta),d\theta )

Partial Fractions

• Linear: ( \frac{A}{x - r} )
• Irreducible quad: ( \frac{Ax + B}{x^2 + bx + c} )
• Example: ( \int \frac{6x^3 + 19x^2 + 12x + 5}{(x+1)^3(x^2+1)} dx )

2. Applications of Integration

Area Between Curves

• Horizontal slices: ( \int (f(x) - g(x)) dx )
• Vertical slices: ( \int (f(y) - g(y)) dy )
• Example: Region bounded by ( y = \sqrt{x},\ x = 9,\ y = 0 )

Volumes of Revolution

Disk/Washer Method

• About x-axis: ( \pi\int_{a}^{b} (R(x)^2 - r(x)^2) dx )
• About y-axis: ( \pi\int_{c}^{d} (R(y)^2 - r(y)^2) dy )
• Example: Revolving ( y = \sqrt{x} ) around x- and y-axes

Shell Method

• About y-axis: ( 2\pi\int_{a}^{b} x\cdot f(x) dx )
• About x-axis: ( 2\pi\int_{c}^{d} y\cdot g(y) dy )
• Example: Volume by revolving ( y = x^2 + 1 ) about y-axis

Work

• Pumping liquids: ( W = \rho g \int_{a}^{b} (\text{height to lift}) \cdot A(y) dy )
• Example: Inverted cone with height 7 m and radius 2 m_

3. Sequences & Series

Definitions

• Sequence: ( a_n )
• Series: ( \sum a_n )
• Geometric: ( a, ar, ar^2, \ldots \Rightarrow S = \frac{a}{1 - r} \text{ if } |r| < 1 )
• Example: ( 0.52323\ldots ) as a geometric series

Convergence Tests

• Divergence Test: If ( \lim a_n \neq 0 ), then ( \sum a_n ) diverges
  • Example: ( \sum \frac{k^6}{\ln k} )
• p-series: ( \sum \frac{1}{n^p} ) converges if ( p > 1 )
• Comparison/Limit Comparison: Compare to known series
  • Example: ( \sum \frac{n^2}{n^3 - \ln(n)} )
• Ratio/Root Test: Use for power series and factorials
  • Example: ( \sum \frac{22^n \sqrt{n}}{n!} )

Absolute vs Conditional Convergence

• Absolute: ( \sum |a_n| ) converges
• Conditional: ( \sum a_n ) converges, but ( \sum |a_n| ) does not
• Example: Compare Series 1 and 2: ( \sum \frac{(-1)^n(n+1)}{2n^2 + 5} )

4. Power & Taylor Series

Common Power Series

• ( \frac{1}{1 - x} = \sum x^n )
• ( e^x = \sum \frac{x^n}{n!} )
• ( \sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!} )
• ( \cos x = \sum \frac{(-1)^n x^{2n}}{(2n)!} )
• ( \ln(1 + x) = \sum \frac{(-1)^n x^{n+1}}{n+1} )
• Example: Maclaurin of ( \frac{x^5}{3 - 2x^2} )

Taylor Series

• ( f(x) = \sum \frac{f^{(n)}(a)}{n!}(x - a)^n )
• Example: ( f(x) = 8 - 2x - 3x^2 + 2x^3 ) about ( x_0 = 1 )

Interval of Convergence

• Use Ratio or Root Test, then test endpoints separately
• Example: ( \sum \frac{(-1)^n(x+3)^n}{\sqrt{n}} )

5. Parametric & Polar Curves

Parametric Equations

• ( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} )
• Arc Length: ( \int_{a}^{b} \sqrt{(dx/dt)^2 + (dy/dt)^2},dt )
• Example: ( x = 2t^2 + 1,\ y = t^3 - t^2 + t - 1 )_

Polar Coordinates

• Area: ( A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 d\theta )
• Arc Length: ( \int \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta )
• Convert:
  • ( x = r \cos \theta,\ y = r \sin \theta )
  • ( r^2 = x^2 + y^2,\ \tan \theta = y/x )
• Example: Area of inner loop of ( r = 1 + 2 \sin(\theta) )_