5.1 Approximating Areas

The use of sigma (summation) notation of the form $\sum_{i = 1}^{n} a_{i}$ is useful for expressing long sums of values in compact form.
For a continuous function defined over an interval $[a, b],$ the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
The width of each rectangle is $Δ x = \frac{b - a}{n} .$
Riemann sums are expressions of the form $\sum_{i = 1}^{n} f (x_{i}^{*}) Δ x,$ and can be used to estimate the area under the curve $y = f (x) .$ Left- and right-endpoint approximations are special kinds of Riemann sums where the values of ${x_{i}^{*}}$ are chosen to be the left or right endpoints of the subintervals, respectively.
Riemann sums allow for much flexibility in choosing the set of points ${x_{i}^{*}}$ at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

5.2 The Definite Integral

The definite integral can be used to calculate net signed area, which is the area above the x-axis less the area below the x-axis. Net signed area can be positive, negative, or zero.
The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
The properties of definite integrals can be used to evaluate integrals.
The area under the curve of many functions can be calculated using geometric formulas.
The average value of a function can be calculated using definite integrals.

The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value c such that $f (c)$ equals the average value of the function. See The Mean Value Theorem for Integrals.
The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See Fundamental Theorem of Calculus, Part 1.
The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See The Fundamental Theorem of Calculus, Part 2.

The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero.
The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.

Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable u and du for appropriate expressions in the integrand.
When using substitution for a definite integral, we also have to change the limits of integration.

Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.
Substitution is often used to evaluate integrals involving exponential functions or logarithms.

Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions lead directly to integration formulas involving inverse trigonometric functions.
Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.
Substitution is often required to put the integrand in the correct form.