Flashcards in 9.4.6 Evaluating Definite Integrals Deck (13):
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.
- 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
Evaluate.3∫1 3x dx
Evaluate.2∫1 ex dx
e (e − 1)
Evaluate.3∫1 3x^2 dx
Evaluate the given definite integral.
∫π/2 0 sinx dx
Evaluate. 3 ∫ 0 x dx
Evaluate.a ∫ 0 x^2 dx
Evaluate.a ∫−a 3 dx
Evaluate.3∫1 (2x+5) dx
Evaluate.4∫0 3 dx