Unit Four: Integrals Flashcards
What is the indefinite integration of {0dx?
C
What is the indefinite integration of {kdx?
Kx + c
What is the indefinite integration of {k f(x) dx?
K {f(x) dx
What is the indefinite integration of {f(x)+/-g(x)dx?
{f(x)dx +/- {g(x)dx
What is the indefinite integration of {x^ndx?
1/(n+1) x^(n+1) + c
What is the summation of c if if i starts at 1 and goes to n?
c * n
What is the summation of i if if i starts at 1 and goes to n?
n*(n+1) divided by 2
What is the summation of i^2 if if i starts at 1 and goes to n?
n(n+1)(2n+1) divided by 6
What is the summation of i^3 if if i starts at 1 and goes to n?
n^2*(n+1)^2 divided by 4
How do you find the area under a curve (sums)?
Find the infinite sum of rectangles infinitely small in width
How do you change a sum of infinite rectangles into an integration?
Integration from first x to second x of the function
If f is defined at a, what is the integration from a to a of a function?
0, there is no width
If f is defined on [a,b] then what is the relationship between the integration from b to a of a function and the integration from a to b of the same function?
The second is negative of the first
How do you integrate an absolute value?
Break it apart into two integrations with two different functions
What is the fundamental theorem of calculus? MEMORIZE
If a function f is continuous on [a,b] and F is an antiderivative of f on [a,b], then the integration of f(x) from a to b is equal to F(b)-F(a)
What is the mean value theorem for integrals? MEMORIZE
If f is continuous on [a,b], then there exists a number c in [a,b] such that the the integration of f(x) from a to b is equal to f(c)(b-a)
What is the average value of a function?
If f is integratabtle on [a,b], then the average value of f on the interval is 1/(b-a) times the integration of f(x) from a to b
What is the second fundamental theorem of calculus? MEMORIZE
If f is continuous on an open interval I containing a, then for every x in I the derivative of the integration of f(t) from a to x is f(x)
What is the integration of f(g(x))g’(x)dx?
*It is equal to F(g(x))+C
If u is equal to g(x), then du is equal to g’(x)dx then the integration of f(u)du is F(u) + C
What is the integration of f(g(x))g’(x)dx from a to b?
If u is equal to g(x), then du is equal to g’(x)dx then it is equal to the integration of f(u)du from g(a) to g(b)
What is the formula for trapezoidal approximations of the area under a curve?
(b-a)/n times 1/2 times (f(x0) + 2f(x1) +2f(x2) + … + 2f(Xn-1) + f(xn))
What is the formula for Simpson’s rule and what is the requirement?
(b-a)/n times 1/3 times (f(x0) + 4f(x1) +2f(x2) + 4f(x3) + 2f(x4) + … + 4f(Xn-1) + f(xn)), n is even
Derivative of sin
Cos
Derivative of cos
-sin
Derivative of tan
Sec^2
Derivative of csc
-csc*cot
Derivative of sec
Sec*Tan
Derivative of cot
-csc^2
