6.4 Properties Of Definite Integralsap Calculus

1. 6.4 Properties Of Definite Integralsap Calculus Calculator
2. 6.4 Properties Of Definite Integralsap Calculus Solver

6.4 Properties Of Definite Integralsap Calculus Calculator

6.4 Properties Of Definite Integralsap Calculus Solver

World of warships game manual. 6.4: Physics Applications - Work, Force, and Pressure While there are many different formulas that we use in solving problems involving work, force, and pressure, it is important to understand that the fundamental ideas behind these problems are similar to several others that we've encountered in applications of the definite integral. Mdb explorer mac mdb explorer for mac os. FUN-6.A.2 Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals. FUN-6.A.3 The definition of the definite integral may be extended to functions with removable or jump discontinuities.

1. 6.4 Properties Of Definite Integralsap Calculus Calculator
2. 6.4 Properties Of Definite Integralsap Calculus Solver

6.4 Properties Of Definite Integralsap Calculus Calculator

6.4 Properties Of Definite Integralsap Calculus Solver

World of warships game manual. 6.4: Physics Applications - Work, Force, and Pressure While there are many different formulas that we use in solving problems involving work, force, and pressure, it is important to understand that the fundamental ideas behind these problems are similar to several others that we've encountered in applications of the definite integral. Mdb explorer mac mdb explorer for mac os. FUN-6.A.2 Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals. FUN-6.A.3 The definition of the definite integral may be extended to functions with removable or jump discontinuities.

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).)