Numerical Integration Flashcards
What does the midpoint rule use?
Midpoint rule uses rectangles of the same width
What is the width of each rectangle/trapezium?
xn - xo / n
What is the formula for the area of a rectangle using the midpoint rule?
(xn - xo / n) * (y1/2 + y3/2 + y5/2 + … + yn-1/2)
How do you improve the accuracy of the estimate?
Using more strips
What is the trapezium rule?
Trapeziums of the same width estimate area under curve
What is the formula for the area using the trapezium rule?
h/2 (y1 + 2(y2 + … + yn-1) + yn)
How do you know if the area is an under/over estimate?
Underestimate is when the curve is concave down
Overestimate is when the curve is concave up
What is the formula for T2n?
T2n = Tn + Mn / 2
What is the relationship of T n and M n as an estimate of the integral?
Tn and M n are converse in that one is an underestimate and the other is an overestimate
What is the Simpson’s rule?
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
What is the formula for the width of each piece?
h = b - a / 2n
What is the notation for Simpson’s Rule?
S 2 use one quadratic
S 4 use two quadratics
What is the formula for S2n?
S2n = h/3 (y0 + y2 + 4(yodd) + 2(yeven))
What is the formula for the Simpson’s Rule using the midpoint and trapezium method?
S2n = 2Mn + Tn / 3
What is the formula for the Simpson’s Rule using the trapezium method?
S2n = 4T2n - Tn / 3