How do you write the Simpson’s rule and Trapezoid rule approximations to the #intsinx/x# over the inteval [0,1] with #n=6#?

Answer 1
Simpson's sule with n=6 is simply: #int_0^1f(x)=1/18[f(0)+4f(1/6)+2f(2/6)+4f(3/6)+2f(4/6)+4f(5/6)+f(1)]#
Here f(x) is just #sinx/x# and so one must calculate this for x=0, 1/6, ... 5/6, 1 and put the results into the formula.
The Trapezoidal rule is similar: #int_0^1f(x)=1/12[f(0)+2f(1/6)+2f(2/6)+2f(3/6)+2f(4/6)+2f(5/6)+f(1)]#
Note that, although #f(0)# does not exist, we know that #lim_(xrarr0)f(x)=lim_(xrarr0)sinx/x=1#.
Use #1# where the formula calls for #f(0)#.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

Simpson's Rule: [ \text{Approximation using Simpson's Rule} = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6) \right] ]

Trapezoid Rule: [ \text{Approximation using Trapezoid Rule} = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6) \right] ]

Given (n = 6), the interval ([0,1]) is divided into 6 equal subintervals, hence (\Delta x = \frac{1-0}{6} = \frac{1}{6}). Therefore, (x_0 = 0), (x_1 = \frac{1}{6}), (x_2 = \frac{1}{3}), (x_3 = \frac{1}{2}), (x_4 = \frac{2}{3}), (x_5 = \frac{5}{6}), and (x_6 = 1).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7