Study guide
Integration reverses differentiation and reframes it as accumulation: adding up infinitely many infinitesimal pieces to find a total. This final chapter covers antiderivatives, the Fundamental Theorem of Calculus that links derivatives and integrals, differential equations that model change over time, and the geometric applications — area and volume — that make accumulation tangible.
Antiderivatives and Definite Integrals
An antiderivative of f is any function F such that F'(x) = f(x). Because the derivative of a constant is zero, antiderivatives come in families differing by an arbitrary constant C, written as the indefinite integral, integral of f(x) dx = F(x) + C. Basic antiderivative rules mirror derivative rules in reverse: the power rule for integration states that the integral of x^n dx equals x^(n+1)/(n+1) + C for n not equal to -1, the integral of 1/x dx is ln|x| + C, the integral of e^x dx is e^x + C, and the integral of cos(x) dx is sin(x) + C. For the polynomial 3x^2 - 4x + 1, the antiderivative is x^3 - 2x^2 + x + C. A definite integral, written integral from a to b of f(x) dx, represents the signed area between the curve and the x-axis over [a, b], and it evaluates to a single number rather than a family of functions — any region below the x-axis contributes negatively. Definite integrals can also be approximated numerically using Riemann sums (left, right, or midpoint rectangles) or the trapezoidal rule when only a table of values is given rather than a formula, which is common on calculator-active questions using unevenly spaced data.
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) has two parts that together connect differentiation and integration as inverse processes. (Be aware that textbooks disagree on which part is numbered first — many, including Stewart, call the accumulation-function statement Part 1 and the evaluation statement Part 2, the reverse of the labeling used here — so learn both statements rather than the labels.) One part, the evaluation statement, says that if F is any antiderivative of a continuous function f, then integral from a to b of f(x) dx equals F(b) - F(a). For integral from 0 to 2 of (x^2 + 1) dx, an antiderivative is x^3/3 + x, so the definite integral equals (8/3 + 2) - (0 + 0) = 14/3, approximately 4.667. The other part of FTC addresses accumulation functions of the form g(x) = integral from a to x of f(t) dt, and states that g'(x) = f(x) — differentiating an integral with respect to its upper limit simply returns the original integrand evaluated there. For g(x) = integral from 0 to x of sqrt(t^2 + 1) dt, the derivative is g'(x) = sqrt(x^2 + 1) directly, with no further integration required. When the upper limit is itself a function of x, such as integral from 0 to x^2 of f(t) dt, the chain rule applies on top of FTC, multiplying by the derivative of the upper limit: the result is f(x^2) times 2x. Accumulation functions frequently appear as context-based problems describing a rate (such as water flowing into a tank in gallons per minute), where the accumulation function itself represents total amount accumulated by time x.
Separable Differential Equations
A differential equation relates a function to its own derivative, and a separable differential equation is one where the variables can be algebraically sorted onto opposite sides of the equation before integrating. The general method: rewrite dy/dx as a product or quotient of a function of x and a function of y, move all y-terms (including dy) to one side and all x-terms (including dx) to the other, integrate both sides separately, and add a single constant of integration. If an initial condition is given, substitute it immediately afterward to solve for that constant, producing a particular solution rather than a general family. Consider dy/dx = ky (exponential growth or decay), with initial population y(0) = 50 growing to y(3) = 200. Separating gives (1/y) dy = k dx, and integrating both sides yields ln|y| = kx + C, so y = Ae^(kx) where A = e^C. Applying y(0) = 50 gives A = 50. Applying y(3) = 200 gives 200 = 50e^(3k), so 4 = e^(3k), meaning k = ln(4)/3, approximately 0.462. Slope fields, small line segments drawn at grid points representing the value of dy/dx at each point, offer a graphical preview of solution curves without solving the equation algebraically, and exam questions often ask you to sketch a solution curve through a given point by following the slope pattern shown.
Area Between Curves, Volume of Revolution, and Average Value
The area between two curves y = f(x) and y = g(x) from x = a to x = b, where f(x) is greater than or equal to g(x) on that interval, is integral from a to b of [f(x) - g(x)] dx. When the curves cross within the interval, locate the intersection points first (by setting f(x) equal to g(x)) and split the integral there, since the 'top' and 'bottom' functions swap. For y = x^2 and y = x + 2, setting them equal gives x^2 - x - 2 = 0, factoring as (x-2)(x+1) = 0, so the curves intersect at x = -1 and x = 2; since x + 2 lies above x^2 there, the area is integral from -1 to 2 of [(x+2) - x^2] dx, which evaluates to 4.5 square units. The disk method computes the volume of a solid formed by revolving a region around an axis: revolving the region under y = f(x) from x = a to x = b around the x-axis gives volume pi times the integral from a to b of [f(x)]^2 dx, since each cross-section is a circular disk of radius f(x). Revolving y = sqrt(x) from x = 0 to x = 4 around the x-axis gives volume pi times the integral from 0 to 4 of x dx, equal to pi times 8, or 8 pi, approximately 25.13 cubic units. The average value of a continuous function f on [a, b] is (1/(b-a)) times the integral from a to b of f(x) dx; for f(x) = x^2 on [0, 3], the average value is (1/3) times 9 = 3, and the Mean Value Theorem for Integrals guarantees some c in [a, b] where f(c) equals this average value.
Key terms
- Antiderivative
- — A function F such that F'(x) = f(x); antiderivatives of the same f differ only by a constant.
- Indefinite integral
- — The general family of antiderivatives of a function, written integral of f(x) dx = F(x) + C.
- Definite integral
- — The signed area between a curve and the x-axis over an interval [a,b], evaluating to a single numerical value.
- Riemann sum
- — An approximation of a definite integral using the sum of areas of rectangles (left, right, or midpoint) built from subintervals.
- Fundamental Theorem of Calculus, Part 1
- — States that integral from a to b of f(x) dx equals F(b) - F(a), where F is any antiderivative of f (part numbering varies by textbook; many label this statement Part 2).
- Fundamental Theorem of Calculus, Part 2
- — States that the derivative of an accumulation function g(x) = integral from a to x of f(t) dt equals f(x) (part numbering varies by textbook; many label this statement Part 1).
- Separable differential equation
- — A differential equation whose variables can be algebraically separated onto opposite sides before both sides are integrated.
- Slope field
- — A graphical grid of short line segments showing the slope dy/dx at each point, used to visualize solutions to a differential equation.
- Disk method
- — A technique for finding the volume of a solid of revolution by integrating pi times the square of the radius function across an interval.
- Average value of a function
- — The value (1/(b-a)) times the integral from a to b of f(x) dx, representing the height of a rectangle with the same area as the region under the curve.
- Area between curves
- — The definite integral of the difference between an upper function and a lower function over the interval where that ordering holds.
Exam tips
- When a definite integral is described only by a table of unevenly spaced values, use a Riemann sum or trapezoidal approximation rather than assuming you can find an exact antiderivative.
- For accumulation functions with a variable upper limit, apply the chain rule whenever that limit is itself a function of x, not just a bare x.
- Always find and use the correct order of intersection points before splitting an area-between-curves integral where functions cross.
- For separable differential equations, apply the initial condition after integrating both sides, and keep the absolute value inside a natural log until you solve for the constant.
- Double-check whether an average-value problem is asking for the average of f(x) over an interval versus the average rate of change of f, since these use different formulas.