Riemann Sums and Fun of Calc (5.3-5.4)

Question 1

How to find change in x

Answer 1

b-a / n

Question 2

On an interval from Xj to Xj++ the height of rectangle is

Answer 2

f of Xj

Question 3

The riemann sum on a to b

Answer 3

f (X1) change in x + f(X2) change of x + … + f(Xn) change in x

Question 4

How is riemann sum from a to b denoted in summation form

Answer 4

Sigma (top - n and bottom- j=1) f(Xj) change in x

Question 5

Left rule, vs right rule and midpoint

Answer 5

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

Question 6

riemann sums approximate …

Answer 6

the definite integral from a to b

Question 7

riemann sum theorem

Answer 7

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

Question 8

riemann sums theorem equation

Answer 8

limit as n approaches inf of f(X1)changex + f(X2)changex + …+ f(Xn)changex
this = the definite integral from a to b

Question 9

signa (top-n, bottom-i=1) of c

Answer 9

n times c

Question 10

sigma (top-n, bottom-i=m) of a1 + b1

Answer 10

sigma(top-n, bottom-i=m) of a1 + sigma(top-n, bottom-i=m) of b1

Question 11

sigma (top-n, bottom-i=m) of c a1

Answer 11

c times the sigma of a1

Question 12

sigma (top-n, bottom-i=1) of i=1+2+3+…+n =

Answer 12

n(n+1) / 2

Question 13

sigma (top-n, bottom-i=1) of i^2 = 1^2 +2^2+…+n^2 =

Answer 13

n(n+1)(2n+1) / 6

Question 14

sigma (sigma (top-n, bottom-i=1) of i^3 =

Answer 14

n(n+1)/2 all ^2

Question 15

Steps for evaluating using the ugly formal definition of definite integral

Answer 15

1. find change in x
2. Find Xk - k times change in x
3. find f(Xk) times change in x
4. plug in previous answer into sigma
5. set the definite integral given equal to and plug in and find the limit (pass through limit)

Question 16

2 parts of fundamental theorem of calc

Answer 16

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)

Question 17

How to solve definite integral with funky bounds and only a variable

Answer 17

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

Question 18

Theorem of area in between curves

Answer 18

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

Question 19

Integrate from a to c when

Answer 19

there is a switch on which graph is higher

Question 20

How to find where curves intersect

Answer 20

set f and g equal and solve

Question 21

How to solve an area bounded by curves

Answer 21

1. find where intersect
2. make integral with bounds and correct order of f and g
3. solve

Question 22

Average value of f on closed interval a to b

Answer 22

f avg = 1/b-a times integral from a to b of f

Question 23

How do you solve a Riemann sum, estimating

Answer 23

1. find change in x
2. find the points based on left/right/mid rule
3. plug in those points
4. use riemann sum equation of change in x times f of each point added