10/8/2021

6.4 Properties Of Definite Integralsap Calculus

20
Integrands: (f), (g), (u), (v)
Antiderivatives: (F), (G)
Independent variables: (x), (t)
Limits of integration: (a), (b), (c), (d)

9 Use the properties of definite integrals to find f(x) dx for the following function. tex 6 4x + 1 if xs7 f(x) = -0.6x + 7 if x 7 9 f(x) dx = 6 (Simplify your answer.) Get more help from Chegg. Definite Integrals: Definite Integrals are defined as: and also: which is the fundamental theorem of calculus, where a,b. Special Integrals: If f is integrable on a,b and a Properties Of Definite Integrals.

Subintervals of integration: (Delta {x_i})
Arbitrary point of a subinterval: ({xi_i})
Natural numbers: (n), (i)
Area of a curvilinear trapezoid: (S)
Calculus
  1. Let a real function (fleft( x right)) be defined and bounded on the interval (left[ {a,b} right]). Let us divide this interval into (n) subintervals. In each interval, we choose an arbitrary point ({xi_i}) and form the integral sum (sumlimits_{i = 1}^n {fleft( {{xi _i}} right)Delta {x_i}}) where (Delta {x_i}) is the length of the (i)th interval. The definite integral of the function (fleft( x right)) over the interval (left[ {a,b} right]) is defined as the limit of the integral sum (Riemann sums) as the maximum length of the subintervals approaches zero.
    (require{AMSmath.js}{largeintlimits_a^bnormalsize} {fleft( x right)dx} =) (limlimits_{substack{n to inftytext{max},Delta {x_i} to 0}} sumlimits_{i = 1}^n {fleft( {{xi _i}} right)Delta {x_i}} ,) where (Delta {x_i} = {x_i} – {x_{i – 1}},) ({x_{i – 1}} le {xi _i} le {x_i}.)
  2. The definite integral of (1) is equal to the length of the interval of integration:
    ({largeintlimits_a^bnormalsize} {1,dx} = b – a)
  3. A constant factor can be moved across the integral sign:
    ({largeintlimits_a^bnormalsize} {kfleft( x right)dx} =) ( k{largeintlimits_a^bnormalsize} {fleft( x right)dx} )
  4. The definite integral of the sum of two functions is equal to the sum of the integrals of these functions:
    ({largeintlimits_a^bnormalsize} {left[ {fleft( x right) + gleft( x right)} right]dx} =) ( {largeintlimits_a^bnormalsize} {fleft( x right)dx} + {largeintlimits_a^bnormalsize} {gleft( x right)dx} )
  5. The definite integral of the difference of two functions is equal to the difference of the integrals of these functions:
    ({largeintlimits_a^bnormalsize} {left[ {fleft( x right) – gleft( x right)} right]dx} =) ( {largeintlimits_a^bnormalsize} {fleft( x right)dx} – {largeintlimits_a^bnormalsize} {gleft( x right)dx} )
  6. If the upper and lower limits of a definite integral are the same, the integral is zero:
    ({largeintlimits_a^anormalsize} {fleft( x right)dx} = 0)
  7. Reversing the limits of integration changes the sign of the definite integral:
    ({largeintlimits_a^bnormalsize} {fleft( x right)dx} =) ( -{largeintlimits_b^anormalsize} {fleft( x right)dx})
  8. Suppose that a point (c) belongs to the interval (left[ {a,b} right]). Then the definite integral of a function (fleft( x right)) over the interval (left[ {a,b} right]) is equal to the sum of the integrals over the intervals (left[ {a,c} right]) and (left[ {c,b} right]:)
    ({largeintlimits_a^bnormalsize} {fleft( x right)dx} =) ( {largeintlimits_a^cnormalsize} {fleft( x right)dx} ) (+; {largeintlimits_c^bnormalsize} {fleft( x right)dx})
  9. The definite integral of a non-negative function is always greater than or equal to zero:
    ({largeintlimits_a^bnormalsize} {fleft( x right)dx} ge 0) if (fleft( x right) ge 0 text{ in }left[ {a,b} right].)
  10. The definite integral of a non-positive function is always less than or equal to zero:
    ({largeintlimits_a^bnormalsize} {fleft( x right)dx} le 0) if (fleft( x right) le 0 text{ in } left[ {a,b} right].)
  11. Fundamental theorem of calculus
    ({largeintlimits_a^bnormalsize} {fleft( x right)dx} =) ( {left. {Fleft( x right)} right _a^b} =) ( Fleft( b right) – Fleft( a right),) if (F’left( x right) = fleft( x right).)
  12. Substitution rule for definite integrals
    If (x = gleft( t right)), then ({largeintlimits_a^bnormalsize} {fleft( x right)dx} =) ({largeintlimits_c^dnormalsize} {fleft( {gleft( t right)} right)g’left( t right)dt}, ) where (c = {g^{ – 1}}left( a right),) (d = {g^{ – 1}}left( b right).)
  13. Integration by parts for definite integrals
    ({largeintlimits_a^bnormalsize} {udv} =) (left.{left( {uv} right)}right _a^b – {largeintlimits_a^bnormalsize} {vdu} )
  14. Trapezoidal approximation of a definite integral
    ({largeintlimits_a^bnormalsize} {fleft( x right)dx} =) (largefrac{{b – a}}{{2n}}normalsizeBig[ {fleft( {{x_0}} right) + fleft( {{x_n}} right) }) (+;{ 2sumlimits_{i = 1}^{n – 1} {fleft( {{x_i}} right)} } Big])
  15. Approximation of a definite integral using Simpson’s rule
    ({largeintlimits_a^bnormalsize} {fleft( x right)dx} =) ({largefrac{{b – a}}{{3n}}normalsize}Big[ {fleft( {{x_0}} right) + 4fleft( {{x_1}} right) }) (+;{2fleft( {{x_2}} right)}) (+;{ 4fleft( {{x_3}} right) + 2fleft( {{x_4}} right) + ldots}) (+;{4fleft( {{x_{n – 1}}} right) + fleft( {{x_n}} right)} Big],)
    where ({x_i} = a + {largefrac{{b – a}}{n}normalsize} i,) (i = 0,1,2, ldots ,n.)
  16. Area under a curve
    (S = {largeintlimits_a^bnormalsize} {fleft( x right)dx} =) (Fleft( b right) – Fleft( a right),) where (F^{,prime}left( x right) = fleft( x right).)
  17. Area between two curves
    (S = {largeintlimits_a^bnormalsize} {left[ {fleft( x right) – gleft( x right)} right]dx} =) (Fleft( b right) – Gleft( b right) ) (-;Fleft( a right) + Gleft( a right),) where (F^{,prime}left( x right) = fleft( x right)), (G^{,prime}left( x right) = gleft( x right).)

6.4 Properties Of Definite Integralsap Calculus Solver

6.4 properties of definite integralsap calculus integrals

6.4 Properties Of Definite Integralsap Calculus Integrals

6.4 Properties of Definite Integrals Notes 6.4 Prop's of Def. Integrals FRQ 2003 #4 Key Hw 6.4. FRQ 1998 #3 Key FRQ. 6.4 - Shell Method (x-axis). 37 Sophia partners guarantee credit transfer. 300 Institutions have accepted or given pre-approval for credit transfer. The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 32 of Sophia’s online courses. Many different colleges a.

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