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AP Calculus AB: 9.4.6 Evaluating Definite Integrals

Mathematics13 CardsCreated 3 months ago

This content covers methods for evaluating definite integrals, especially using substitution, and explains how changing the limits to the new variable simplifies calculations. It also highlights the geometric interpretation of definite integrals as areas under curves and provides examples to illustrate these concepts.

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Evaluating Definite Integrals

  • When working with integration by substitution and definite integrals, the limits of integration are given in terms of the original variable.

  • Since there is a connection between the definite integral and the area between a curve and the x-axis, some definite integrals can be solved geometrically.

  • Another way to evaluate definite integrals by substitution is to change the limits of integration so that they are in terms of the new variable.

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Key Terms

Term
Definition

Evaluating Definite Integrals

  • When working with integration by substitution and definite integrals, the limits of integration are given in terms of the original variable...

note

  • Definite integrals appear with limits of integration. They produce numerical values for as results. Geometrically a definite integral repre...

Evaluate.2∫−1 4 dx

12

Evaluate.3∫1 3x dx

12

Evaluate.2∫1 ex dx

e (e − 1)

Evaluate.3∫1 3x^2 dx

26

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AP Calculus AB: 9.4.6 Evaluating Definite Integrals
TermDefinition

Evaluating Definite Integrals

  • When working with integration by substitution and definite integrals, the limits of integration are given in terms of the original variable.

  • Since there is a connection between the definite integral and the area between a curve and the x-axis, some definite integrals can be solved geometrically.

  • Another way to evaluate definite integrals by substitution is to change the limits of integration so that they are in terms of the new variable.

note

  • Definite integrals appear with limits of integration. They produce numerical values for as results. Geometrically a definite integral represents the area between the curve described by the integrand and the x-axis.

  • When you evaluate definite integrals by substitution, the limits of integration are x-values, not u-values. One way to avoid this difficulty is to determine the antiderivative using an indefinite integral.

  • Once you have determined the antiderivative, you can
    evaluate the indefinite integral. You do not need the constant of integration C, since it will be cancelled.

  • This integrand does not resemble any of the basic patterns, and the choices for integration by substitution do not seem to simplify the integrand.

  • One way to better understand this integral is to consider it graphically.

  • By setting y equal to the integrand, you can square both sides and arrive at the equation of a circle. Since the limits of integration are x = 0 and x = 1, the region is one quarter of a circle of radius 1.

  • Use the formula for the area of a circle and divide by 4 to arrive at the area of the region. The result is the value of indefinite integral.

Evaluate.2∫−1 4 dx

12

Evaluate.3∫1 3x dx

12

Evaluate.2∫1 ex dx

e (e − 1)

Evaluate.3∫1 3x^2 dx

26

Evaluate the given definite integral.

∫π/2 0 sinx dx


1

Evaluate. 3 ∫ 0 x dx

9/2

Evaluate.a ∫ 0 x^2 dx

a^3/3

Evaluate.a ∫−a 3 dx

6a

Evaluate.3∫1 (2x+5) dx

18

Evaluate.4∫0 3 dx

12

Evaluate.3 ∫ −1 (5−x) dx

16