Calculus II

The definite integral computes the net signed area between a function's graph and the x-axis over an interval. It's built by slicing the region into thin rectangles, summing their areas, and taking the limit as the rectangles become infinitely thin — the Riemann sum construction. The integral is the second pillar of calculus, and the Fundamental Theorem of Calculus reveals it to be intimately connected to the derivative: they are, in a precise sense, inverse operations.

The Fundamental Theorem of Calculus (FTC) is the bridge connecting differentiation and integration, proving they are inverse operations. It has two parts: Part 1 says the derivative of an integral (with variable upper limit) gives back the original function; Part 2 gives a practical method for evaluating definite integrals using antiderivatives — the technique used in essentially every introductory calculus computation.

The arc length formula computes the exact length of a curve y=f(x) over an interval [a,b] by integrating the local stretching factor √(1+[f'(x)]²), which accounts for how much longer a small piece of curve is than its horizontal projection.

An improper integral is a definite integral where either the interval of integration is infinite, or the integrand becomes unbounded somewhere in the interval. Such integrals are defined as a limit of proper (ordinary) integrals, and they may converge to a finite value or diverge.

Integration by partial fractions integrates rational functions (ratios of polynomials) by first decomposing them into a sum of simpler fractions — each with a linear or irreducible quadratic denominator — that can be integrated individually using basic log, arctan, or power rule antiderivatives.

Integration by parts is the integral counterpart of the product rule, used to integrate a product of two functions when no simpler technique applies. It transforms an integral into a (hopefully simpler) different integral, by shifting the derivative from one factor onto the other.

Trigonometric integrals are integrals of products and powers of trigonometric functions, such as ∫sinᵐ(x)cosⁿ(x)dx. They're evaluated using trigonometric identities (Pythagorean, double-angle, and power-reduction formulas) to reduce the integrand to a form solvable by u-substitution.

Trigonometric substitution replaces an algebraic expression involving √(a²-x²), √(a²+x²), or √(x²-a²) with a trigonometric function, exploiting the Pythagorean identities to eliminate the square root and simplify the integral.

u-Substitution (the substitution rule) is the integral counterpart of the chain rule. It simplifies an integral by replacing a complicated expression with a single variable u, transforming the integral into a more recognizable form in terms of u.

Volumes of revolution are solids formed by rotating a two-dimensional region around an axis. Their volumes are computed by integrating cross-sectional areas (disk/washer methods) or cylindrical shell surface areas (shell method) along the axis of rotation.

A Taylor series represents a function as an infinite sum of terms built from its derivatives at a single point. It's one of the most powerful ideas in analysis: a function that might be hard to compute directly (sin, eˣ, ln) can be approximated arbitrarily well by a polynomial, which computers can evaluate quickly using only addition and multiplication.

A parametric curve describes x and y (and possibly z) each as separate functions of a third variable, the parameter t — typically thought of as time. Instead of y = f(x), you write x = x(t), y = y(t). This lets you describe curves that fail the vertical line test (circles, loops, cusps) and naturally encodes motion: position, velocity, and speed all fall out of differentiating the parametric functions.

Differentiating and integrating polar curves r = f(θ) requires adapting Cartesian calculus tools, since x and y are both functions of θ rather than one being a function of the other. The two workhorse formulas are the polar slope formula (for tangent lines) and the polar area formula (for regions swept out by a rotating radius), the latter built from thin circular-sector approximations instead of rectangles.

Polar coordinates locate a point in the plane using a distance r from a fixed origin (the pole) and an angle θ from a fixed reference direction (the polar axis), instead of the (x,y) offsets of Cartesian coordinates. Curves with rotational symmetry — circles centered at the origin, spirals, roses, cardioids — have dramatically simpler equations in polar form than in Cartesian form.

A power series is an infinite series whose terms are powers of (x − a) multiplied by coefficients: Σcₙ(x−a)ⁿ. Unlike a numerical series, a power series defines a function of x, and it converges for x within a specific radius of a and diverges outside it. Power series are the algebraic backbone of Taylor series and let you manipulate transcendental functions (differentiate, integrate, solve differential equations) using term-by-term polynomial-like rules.

Given an infinite series Σaₙ, the central question is: does the sequence of partial sums converge to a finite number, or does it diverge? Directly computing the limit of partial sums is often impossible in closed form, so mathematicians developed a toolbox of convergence tests — the comparison test, ratio test, root test, integral test, alternating series test, and others — each suited to different shapes of terms. Mastering when to reach for which test is one of the most practical skills in calculus.