9.4.6 Evaluating Definite Integrals Flashcards

1
Q

Evaluating Definite Integrals

A
  • 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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2
Q

note

A
  • 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.
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3
Q

Evaluate.2∫−1 4 dx

A

12

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4
Q

Evaluate.3∫1 3x dx

A

12

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5
Q

Evaluate.2∫1 ex dx

A

e (e − 1)

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6
Q

Evaluate.3∫1 3x^2 dx

A

26

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7
Q

Evaluate the given definite integral.

∫π/2 0 sinx dx

A

1

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8
Q

Evaluate. 3 ∫ 0 x dx

A

9/2

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9
Q

Evaluate.a ∫ 0 x^2 dx

A

a^3/3

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10
Q

Evaluate.a ∫−a 3 dx

A

6a

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11
Q

Evaluate.3∫1 (2x+5) dx

A

18

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12
Q

Evaluate.4∫0 3 dx

A

12

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13
Q

Evaluate.3 ∫ −1 (5−x) dx

A

16

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