Numerical integration estimates the value of a definite integral using calculations rather than relying solely on an antiderivative. It is useful when a function is known only through sampled data, when its antiderivative is unavailable or difficult to evaluate, or when an approximation is more practical.
The basic task is to approximate a definite integral such as I=∫abf(x)dx. Geometrically, this integral represents the signed area between the graph of f and the horizontal axis over the interval from a to b.
All flashcards
What is numerical integration?
The use of numerical algorithms to approximate the value of a definite integral.
Why can sampled data motivate numerical integration?
Function values may be available at selected points without a formula for the integrand.
What did quadrature mean historically?
Finding an area by constructing a square with the same area as a given figure.
What area result did Archimedes establish for a parabolic segment?
Its area is four-thirds the area of its inscribed triangle.
What initial-value problem does the accumulated integral F(x)=∫axf(u)du satisfy?
F′(x)=f(x) with F(a)=0.
Quiz questions
How does the midpoint rule approximate an integral over one interval?Answer: It uses a rectangle whose height is the integrand's value at the interval's midpoint.
What geometric approximation underlies the trapezoidal rule on one interval?Answer: A straight line joining the graph's values at the two endpoints.
Which feature characterizes Newton–Cotes quadrature formulas?Answer: They use equally spaced interpolation points.
How does Gaussian quadrature choose its evaluation points?Answer: It selects points and weights rather than requiring equally spaced points.
Where does an adaptive integration method generally add subdivisions?Answer: In regions where the integrand changes rapidly or estimated error is high.