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Calculus Volume 2

Key Concepts

Calculus Volume 2Key Concepts

Key Concepts

1.1 Approximating Areas

  • The use of sigma (summation) notation of the form i=1naii=1nai is useful for expressing long sums of values in compact form.
  • For a continuous function defined over an interval [a,b],[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 Δx=ban.Δx=ban.
  • Riemann sums are expressions of the form i=1nf(xi*)Δx,i=1nf(xi*)Δx, and can be used to estimate the area under the curve y=f(x).y=f(x). Left- and right-endpoint approximations are special kinds of Riemann sums where the values of {xi*}{xi*} 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 {xi*}{xi*} at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

1.2 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.

1.3 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)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.

1.4 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.

1.5 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.

1.6 Integrals Involving Exponential and Logarithmic Functions

  • 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.

1.7 Integrals Resulting in Inverse Trigonometric Functions

  • 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.
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