Numerical Integration Flashcards

1
Q

What does the midpoint rule use?

A

Midpoint rule uses rectangles of the same width

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2
Q

What is the width of each rectangle/trapezium?

A

xn - xo / n

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3
Q

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

A

(xn - xo / n) * (y1/2 + y3/2 + y5/2 + … + yn-1/2)

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4
Q

How do you improve the accuracy of the estimate?

A

Using more strips

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5
Q

What is the trapezium rule?

A

Trapeziums of the same width estimate area under curve

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6
Q

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

A

h/2 (y1 + 2(y2 + … + yn-1) + yn)

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7
Q

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

A

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

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8
Q

What is the formula for T2n?

A

T2n = Tn + Mn / 2

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9
Q

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

A

Tn and M n are converse in that one is an underestimate and the other is an overestimate

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10
Q

What is the Simpson’s rule?

A

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

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11
Q

What is the formula for the width of each piece?

A

h = b - a / 2n

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12
Q

What is the notation for Simpson’s Rule?

A

S 2 use one quadratic
S 4 use two quadratics

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13
Q

What is the formula for S2n?

A

S2n = h/3 (y0 + y2 + 4(yodd) + 2(yeven))

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14
Q

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

A

S2n = 2Mn + Tn / 3

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15
Q

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

A

S2n = 4T2n - Tn / 3

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