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)
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 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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