Riemann Sums and Fun of Calc (5.3-5.4) Flashcards
How to find change in x
b-a / n
On an interval from Xj to Xj++ the height of rectangle is
f of Xj
The riemann sum on a to b
f (X1) change in x + f(X2) change of x + … + f(Xn) change in x
How is riemann sum from a to b denoted in summation form
Sigma (top - n and bottom- j=1) f(Xj) change in x
Left rule, vs right rule and midpoint
left - use left Xj of change in x interval
right- use right Xj of change in x interval
midpoint - use Xj in middle of change in x interval
riemann sums approximate …
the definite integral from a to b
riemann sum theorem
if f(x) is continuous on closed a to b, the limit of the riemann sums (infinitely many little rectangles) when the number of subdivisions goes to inf is the definite integral
riemann sums theorem equation
limit as n approaches inf of f(X1)changex + f(X2)changex + …+ f(Xn)changex
this = the definite integral from a to b
signa (top-n, bottom-i=1) of c
n times c
sigma (top-n, bottom-i=m) of a1 + b1
sigma(top-n, bottom-i=m) of a1 + sigma(top-n, bottom-i=m) of b1
sigma (top-n, bottom-i=m) of c a1
c times the sigma of a1
sigma (top-n, bottom-i=1) of i=1+2+3+…+n =
n(n+1) / 2
sigma (top-n, bottom-i=1) of i^2 = 1^2 +2^2+…+n^2 =
n(n+1)(2n+1) / 6
sigma (sigma (top-n, bottom-i=1) of i^3 =
n(n+1)/2 all ^2
Steps for evaluating using the ugly formal definition of definite integral
- find change in x
- Find Xk - k times change in x
- find f(Xk) times change in x
- plug in previous answer into sigma
- set the definite integral given equal to and plug in and find the limit (pass through limit)
2 parts of fundamental theorem of calc
F(x) = integral from a to b of f(t) => F prime of x = f(x)
integral from a to b of f(x) = F(b) - F(a)
How to solve definite integral with funky bounds and only a variable
plug in intervals a and b into function (like if evaluating) and use chain rule to take derivative, then plug in integral equation into variable and simplify.
derivative of F(g(x)) = F prime of g(x) times g prime
show that you are taking the derivative of it
with only a variable, just switch variable inside integral to that bound variable
Theorem of area in between curves
let f and g be continuous from a to b closed where f > g for all a to b
the area of the region bounded by the curves f and g and the lines a and b
= integral from a to b of f-g
Integrate from a to c when
there is a switch on which graph is higher
How to find where curves intersect
set f and g equal and solve
How to solve an area bounded by curves
- find where intersect
- make integral with bounds and correct order of f and g
- solve
Average value of f on closed interval a to b
f avg = 1/b-a times integral from a to b of f
How do you solve a Riemann sum, estimating
- find change in x
- find the points based on left/right/mid rule
- plug in those points
- use riemann sum equation of change in x times f of each point added
