9.4.6 Evaluating Definite Integrals

Question 1

Evaluating Definite Integrals

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

Evaluate.2∫−1 4 dx

Answer 3

12

Question 4

Evaluate.3∫1 3x dx

Answer 4

12

Question 5

Evaluate.2∫1 ex dx

Answer 5

e (e − 1)

Question 6

Evaluate.3∫1 3x^2 dx

Answer 6

26

Question 7

Evaluate the given definite integral.

∫π/2 0 sinx dx

Answer 7

1

Question 8

Evaluate. 3 ∫ 0 x dx

Answer 8

9/2

Question 9

Evaluate.a ∫ 0 x^2 dx

Answer 9

a^3/3

Question 10

Evaluate.a ∫−a 3 dx

Answer 10

6a

Question 11

Evaluate.3∫1 (2x+5) dx

Answer 11

18

Question 12

Evaluate.4∫0 3 dx

Answer 12

12

Question 13

Evaluate.3 ∫ −1 (5−x) dx

Answer 13

16