Chapter 5 (intro to integration) Flashcards
delta x [a,b]
(b-a)/n
xk
k is any number, Xk= a+k(delta x)
what are the types of rhiemans sums
left (when Xk* is the left limit of the rectangle)
right (when Xk* is the right limit of the rectangle)
middle (when Xk* is the middle limit of the rectangle)
sigma c (c=real number)
cn
sigma k
(n(n=1))/2
sigma k^2
(n(n+1)(2n+1))/6
sigma K^3
(n^2(n+1)^2)/4
xk* left rhiemans sum
a+(k-1)delta x
xk* right rhiemans sum
a+k(delta x)
xk* mid rhiemans sum
a+(k-.5)(delta x)
what is xk*
a point in the Kth interval
reverse
int(a-b) fxdx =-int(b-a) fxdx
general summation rhiemans sum
sigma f(Xk*)(delta x)
steps for evaluating an integral with Riemann sums
- writhe the general equation
- find delta x
- find Xk* using the type of limit you want
- plug in to get one summation equation
- Use summation notation to replace all of the k’s and numbers with n (hint (Xk*)&(delta x) will have parts that pull through)
- simplify the equations until you can take the limit as the system approaches infinity.
area with the fundamental theorem of calculus.
just take the integral and evaluate at the top bound and subtract eh evaluation at the bottom bound.
what trig functions are even
cos x and sec x
what trig functions are odd
everything but cos x and sec x
average value of a function
Af=(1/b-a)*(int a to b) f(x)
mean value theorem for integrals.
if a function is continuous and integratable on the interval a to b then there exist a point on the interval a to b that is equal to the average of the integral from the average value theorem
how to integrate by substitutiom
identify the inside and the outside function
inside function is u
derive u and factor it out of the original function. replace with du
then integrate
plug u back in and go on your merry way
