Chapter 5 | Integration
619
• 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 = b − a n . • Riemann sums are expressions of the form ∑ 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
⎫ ⎭ ⎬ at which the function is evaluated,
i *
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. 5.3 The Fundamental Theorem of Calculus • 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 . 5.4 Integration Formulas and the Net Change Theorem • 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. 5.5 Substitution • 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.
Made with FlippingBook - professional solution for displaying marketing and sales documents online