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Volume Estimates of the Heart through the Use of Simpson's Rule

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Table of Contents

1. Introduction and Objectives 3
2. Simpson’s rule 3
2.1 First proof of Simpson’s rule 4
2.2 Second proof of Simpson’s rule 6
2.3 Error in Simpson’s rule 7
2.4 Number of slices for the approximation to be exact up to a certain number 7
3. Application of the Simpson’s rule to measuring the volume of the heart 8
3.3.2 Sample calculation. 10
4. Conclusion 11
5. Bibliography 12
6. Appendix 12

Introduction and Objectives
I was looking at a program on discovery channel where there were treasure hunter ships.They would scan the seabed it. I noticed then when the seabed came up on the computer screen it was imaged one part at a time, this made me think of integration with the trapezoidal rule where separate …show more content…

Finally, I will draw the main conclusions from this project.
Simpson’s rule
A way that we can see the trapezoidal (or midpoint) rule is that we approximate a function using a first degree polynomial. Following this line of thinking we can say that by using Simpson’s rule we approximate a function using a second degree polynomial. In the graph here above f(x) represents the function that we are trying to approximate
P(x) is the second degree polynomial that we use to approximate the function
The Simpson’s rule can be derived in various ways described in the two following subsections.
First proof of Simpson’s rule
(based on the proof in the book Calculus, Larson Hostetler Edwards’s fifth edition)

∫_a^b▒█(f(x)dx=@ ) ∫_a^b▒〖(A〗 x^2+Bx+C)dx
Integral is taken
∫_a^b▒█(f(x)dx=@ ) [(Ax^3)/3+(Bx^2)/2+C] x = b - a
∫_a^b▒█(f(x)dx=[(A(b^3-a^3))/3+(B〖(b〗^2-a^2))/2+C(b-a)]@ )
By inverse distribution
∫_a^b▒f(x)dx=((b-a)/6)[2A(a^2+ab+b^2 )+3B(b+a)+6C]
∫_a^b▒f(x)dx=((b-a)/6)[(Aa^2+Ba+C)+4[A((b+a)/2)^2+B((b+a)/2)+C]+ (Ab^2+Bb+C)]
∫_a^b▒f(x)dx=((b-a)/6)[f(a)+4f((b+a)/2)+f(b)]
To use Simpson’s rule you divide the interval into n equal subintervals each of width ∆x=(b-a)/n. N is required to be even because for Simpson’s rule we use coordinates to draw a polynomial of the least degree to pass through. For a polynomial to be determined we need three coordinates, we reuse the last coordinate of the previous polynomial to create our next one.

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