Calculus 2: Integration Techniques

• Calculus 2 is a second semester calculus course, building upon the foundations of Calculus 1
• The course typically covers advanced integration techniques, applications of integrals, sequences and series, and parametric equations and polar coordinates

Integration Techniques

• Integration techniques are methods used to find antiderivatives of more complex functions
• These techniques include u-substitution, integration by parts, trigonometric integrals, trigonometric substitution, partial fraction decomposition, and improper integrals
• U-substitution (or substitution) is the reverse of the chain rule, used when the integrand contains a function and its derivative (or a constant multiple thereof)
• Integration by parts is based on the product rule for differentiation and is useful for integrating products of functions, ∫ u dv = uv - ∫ v du
• Trigonometric integrals involve integrating various combinations of trigonometric functions, often requiring trigonometric identities to simplify the integrand
• Trigonometric substitution involves substituting trigonometric functions for expressions involving square roots, which often appears in integrals
• Partial fraction decomposition is used to integrate rational functions by breaking them down into simpler fractions

Applications of Integrals

• Applications of integrals extend the concept of finding areas under curves to solving a variety of problems
• These applications include finding volumes of solids of revolution (disk, washer, and shell methods), arc length of a curve, surface area of a solid of revolution, average value of a function, and applications in physics and engineering such as work, fluid force, and center of mass
• Volumes of solids of revolution are calculated by rotating a region around an axis and integrating the resulting cross-sectional areas
• Arc length is the length of a curve, calculated by integrating the square root of 1 + (dy/dx)^2 (or 1 + (dx/dy)^2) over the interval of interest
• Surface area of a solid of revolution is found by integrating the circumference of cross-sections along the curve that is rotated, 2π∫r ds
• The average value of a function f(x) over an interval [a, b] is given by (1/(b-a))∫[a,b] f(x) dx
• Integrals can be used to calculate work done by a variable force, fluid force on submerged objects, and the center of mass of a region or solid

Sequences and Series

• Sequences are ordered lists of numbers, while series are the sums of the terms in a sequence
• Topics include convergence and divergence of sequences and series, limit of a sequence, geometric series, telescoping series, harmonic series, alternating series, and various convergence tests (e.g., integral test, comparison test, limit comparison test, ratio test, root test)
• A sequence converges if its terms approach a finite limit as n approaches infinity; otherwise, it diverges
• A series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges
• A geometric series has the form Σ ar^n; it converges if |r| < 1 and diverges if |r| ≥ 1
• The integral test relates the convergence of a series to the convergence of an improper integral, useful when the terms of the series are similar to a continuous function
• Comparison tests (direct comparison and limit comparison) compare a given series to a known convergent or divergent series to determine its convergence
• The ratio test and root test use limits involving ratios of successive terms to determine convergence, particularly useful for series involving factorials or exponents
• Alternating series have terms that alternate in sign and may converge even if the absolute values of the terms do not converge

Power Series

• Power series are series of the form Σ c_n(x-a)^n, where c_n are coefficients, x is a variable, and a is the center of the series
• Topics include radius and interval of convergence, differentiation and integration of power series, Taylor series, and Maclaurin series
• The radius of convergence R is a non-negative real number or ∞ such that the power series converges if |x-a| < R and diverges if |x-a| > R
• The interval of convergence is the interval of x-values for which the power series converges; it may include or exclude the endpoints a - R and a + R
• Power series can be differentiated and integrated term-by-term within their interval of convergence, resulting in new power series representing the derivative or integral of the original function
• Taylor series represent a function as an infinite sum of terms involving its derivatives at a single point, f(x) = Σ f^(n)(a) / n!^n
• Maclaurin series are Taylor series centered at a = 0, f(x) = Σ [f^(n)(0) / n!]x^n
• Common Taylor series include those for e^x, sin(x), cos(x), and (1+x)^k

Parametric Equations and Polar Coordinates

• Parametric equations define x and y coordinates as functions of a third variable, called a parameter (usually t), x = f(t), y = g(t)
• Topics include graphing parametric curves, finding derivatives and arc length in parametric form, polar coordinates, polar curves, area in polar coordinates, and relationships between polar and Cartesian coordinates
• Derivatives in parametric form are found using dy/dx = (dy/dt) / (dx/dt)
• Arc length of a parametric curve is given by ∫√((dx/dt)^2 + (dy/dt)^2) dt
• Polar coordinates represent points in the plane using a distance r from the origin and an angle θ from the positive x-axis, (r, θ)
• Polar curves are graphs of equations in polar coordinates, r = f(θ)
• Area in polar coordinates is found by integrating (1/2)r^2 dθ over the appropriate interval of θ values
• The relationships between polar and Cartesian coordinates are x = r cos(θ) and y = r sin(θ)