Numerical Integration

Question 1

What does the midpoint rule use?

Answer 1

Midpoint rule uses rectangles of the same width

Question 2

What is the width of each rectangle/trapezium?

Answer 2

x_n - x_o / n

Question 3

What is the formula for the area of a rectangle using the midpoint rule?

Answer 3

(x_n - x_o / n) * (y_1/2 + y_3/2 + y_5/2 + … + y_n-1/2)

Question 4

How do you improve the accuracy of the estimate?

Answer 4

Using more strips

Question 5

What is the trapezium rule?

Answer 5

Trapeziums of the same width estimate area under curve

Question 6

What is the formula for the area using the trapezium rule?

Answer 6

h/2 (y₀ + 2(y₁ + … + y_n-1) + y_n)

Question 7

How do you know if the area is an under/over estimate?

Answer 7

Underestimate is when the curve is concave down
Overestimate is when the curve is concave up

Question 8

What is the formula for T_2n?

Answer 8

T_2n = T_n + M_n / 2

Question 9

What is the relationship of T _n and M _n as an estimate of the integral?

Answer 9

T_n and M _n are converse in that one is an underestimate and the other is an overestimate

Question 10

What is the Simpson’s rule?

Answer 10

Simpson’s rule involves fitting quadratic functions to points on a curve and using the area under these functions to estimate area under a curve

Question 11

What is the formula for the width of each piece?

Answer 11

h = b - a / 2n

Question 12

What is the notation for Simpson’s Rule?

Answer 12

S ₂ use one quadratic
S ₄ use two quadratics

Question 13

What is the formula for S_2n?

Answer 13

S_2n = h/3 (y₀ + y_2n + 4(y_odd + … + y_2n-1) + 2(y_even + … + y_2n-2))

Question 14

What is the formula for the Simpson’s Rule using the midpoint and trapezium method?

Answer 14

S_2n = 2M_n + T_n / 3

Question 15

What is the formula for the Simpson’s Rule using the trapezium method?

Answer 15

S_2n = 4T_2n - T_n / 3