M459 Integrals Review

A definite integral is a mathematical concept that represents the signed area under a curve defined by a function over a specified interval [a, b]. It is denoted as ∫_a^b f(x) dx.

The Fundamental Theorem of Calculus states that if F is an antiderivative of f on an interval [a, b], then ∫_a^b f(x) dx = F(b) - F(a).

A Riemann sum is a method for approximating the total area under a curve by dividing the region into rectangles, calculating the area of each rectangle, and summing these areas.

Left Riemann sums use the left endpoints of sub-intervals to determine the height of rectangles, while right Riemann sums use the right endpoints.

To calculate the area under a curve using a left Riemann sum, divide the interval into n sub-intervals, determine the height of the function at the left endpoint of each sub-interval, multiply by the width of the sub-intervals, and sum the areas of the rectangles.

The area of a rectangle is calculated as A = width × height.

The limits of integration, a and b, define the interval over which the area under the curve is calculated in a definite integral.

Integration and differentiation are inverse processes; the derivative of an integral function gives the original function, and the integral of a derivative returns the original function (up to a constant).

The trapezoidal rule is a numerical method for estimating the value of a definite integral by approximating the area under the curve as a series of trapezoids.

To evaluate a definite integral using substitution, choose a substitution that simplifies the integral, change the limits of integration accordingly, and then integrate the new function.