| Term | Definition |
|---|---|
| accumulation of change | The total amount of change in a quantity over an interval, represented by the area between a rate of change function and the x-axis. |
| area under a curve | The region between the graph of a function and the x-axis over a specified interval, which represents the accumulation of change when the function is a rate of change. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| completing the square | An algebraic technique for rearranging quadratic expressions into perfect square form to simplify integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| integrands | The function being integrated in an integral expression. |
| long division | An algebraic technique for dividing polynomials to rearrange rational functions into equivalent forms suitable for integration. |
| substitution | An integration technique where a variable is replaced with another expression to simplify the integrand into a more manageable form. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| integrands | The function being integrated in an integral expression. |
| integration by parts | A technique for finding antiderivatives of products of functions, based on the product rule for derivatives. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| linear factors | First-degree polynomial expressions of the form (ax + b) used as denominators in partial fraction decomposition. |
| linear partial fractions | A decomposition technique that expresses a rational function as a sum of simpler fractions with linear denominators, used to simplify integration. |
| nonrepeating factors | Linear factors that appear only once in the denominator of a rational function, used in partial fraction decomposition. |
| rational function | A function expressed as the ratio of two polynomial functions. |
| Term | Definition |
|---|---|
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| improper integral | An integral with one or both limits infinite, or with an unbounded integrand in the interval of integration. |
| infinite limit | Limits that describe the behavior of a function as it approaches infinity or negative infinity, or as the function values grow without bound. |
| limits of definite integrals | A method for evaluating improper integrals by expressing them as limits of definite integrals with finite bounds. |
| unbounded integrand | A function that approaches infinity at one or more points within the interval of integration. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| left Riemann sum | An approximation method for a definite integral using rectangles whose heights are determined by the function values at the left endpoints of subintervals. |
| midpoint Riemann sum | An approximation method for a definite integral using rectangles whose heights are determined by the function values at the midpoints of subintervals. |
| nonuniform partition | A division of an interval into subintervals of varying widths. |
| numerical methods | Computational techniques used to approximate definite integrals when exact analytical solutions are difficult or impossible to obtain. |
| overestimate | An approximation that is greater than the actual value of a function. |
| right Riemann sum | An approximation method for a definite integral using rectangles whose heights are determined by the function values at the right endpoints of subintervals. |
| trapezoidal sum | An approximation method for a definite integral using trapezoids to estimate the area under a curve. |
| underestimate | An approximation that is less than the actual value of a function. |
| uniform partition | A division of an interval into subintervals of equal width. |
| Term | Definition |
|---|---|
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| limiting case | The value or behavior that a mathematical expression approaches as a parameter (such as the width of subintervals) approaches zero. |
| partition | A division of an interval into subintervals used to construct a Riemann sum. |
| Riemann sum | A sum of the form ∑f(x_i*)Δx_i used to approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles. |
| subinterval | One of the smaller intervals created by dividing a larger interval [a,b] into n parts. |
| Term | Definition |
|---|---|
| accumulation function | Functions that represent the accumulated total of a quantity over an interval, typically defined as g(x) = ∫[a to x] f(t) dt. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| accumulation function | Functions that represent the accumulated total of a quantity over an interval, typically defined as g(x) = ∫[a to x] f(t) dt. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| area | In the context of definite integrals, the region between a curve and the x-axis over a specified interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| integral of a constant times a function | The property stating that the integral of a constant multiplied by a function equals the constant times the integral of the function. |
| integral of a function over adjacent intervals | The property stating that the integral of a function over a combined interval equals the sum of integrals over each subinterval. |
| integral of the sum of two functions | The property stating that the integral of a sum of functions equals the sum of the integrals of the individual functions. |
| jump discontinuity | A type of discontinuity where the left-hand and right-hand limits of a function exist but are not equal, causing the function to jump from one value to another. |
| properties of definite integrals | Rules that govern how definite integrals behave, including linearity, reversal of limits, and additivity over adjacent intervals. |
| removable discontinuity | A discontinuity that can be eliminated by defining or redefining the function value at that point to equal the limit. |
| reversal of limits of integration | The property stating that reversing the upper and lower limits of a definite integral changes the sign of the result. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Fundamental Theorem of Calculus | The central theorem linking differentiation and integration, stating that if f is continuous on [a, b] and F is an antiderivative of f, then ∫(a to b) f(x) dx = F(b) - F(a). |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| closed-form antiderivative | An antiderivative that can be expressed using elementary functions and standard mathematical operations. |
| constant of integration | The arbitrary constant C added to an antiderivative to represent the family of all possible antiderivatives of a function. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| integrands | The function being integrated in an integral expression. |
| limits of integration | The upper and lower bounds of a definite integral that must be adjusted when using substitution of variables. |
| substitution of variables | A technique for finding antiderivatives by replacing a variable or expression with a new variable to simplify the integrand. |