Numerical integration

Question 1

how does midpoint rule work

Answer 1

split into segments then make rectangles with height of midpoint then work out total area
create a table with x values, mid x values then y values

Question 2

midpoint rule formula

Answer 2

a-b/n (y1/2 + y3/2…. + y n-1/2)

Question 3

how does trapezium rule work

Answer 3

split into strips then fit a trapezium into each with start and end point then work out total area
make a table of x endpoint values and y values

Question 4

trapezium rule formula

Answer 4

1/2 (a + b/n) ((y0 + yn + 2(y1 + y2….. + y n-1))

Question 5

Combining midpoint and trapezium

Answer 5

T 2n is the average of Tn and Mn
T2 = T1 + M1 /2

Question 6

error bounds of midpoint and trapezium

Answer 6

for a concave up curve, trapezium overestimate and midpoint under
vice versa for concave down
if fully concave up then set trapezium as upper bound and midpoint lower and vice versa
if there is a mix of both then less able to do so

Question 7

How does simpsons rule work

Answer 7

Fitting one or more quadratics to a curve, if one quadratic is fitted it is S2, if 2(i.e two parts of 2 different quadratics) then S4 and so on

Question 8

Simpsons rule formulae

Answer 8

S 2n = 2Mn + Tn / 3
S 2n = 4T2n - Tn / 3