Question

I am having trouble with this pointless Simpson's Rule...-_-

Integral of (9-x^2) from 3 to 5.

Please help me out!

Answer 1

\[\int\limits_{3}^{5} 9-x ^{2}=[9x-\frac{x ^3}{3} ]_{3}^{5} \]\[= 9(3)-\frac{3^3}{3}-9(5)+\frac{3^5}{3} =27-9-45+81\]\[=54\]

Answer 2

Thank you, however, I was having trouble understanding how to solve it using Simpson's Rule.

Answer 3

All of my values were coming out negative. =(

Answer 4

That's because the answer is -44/3

Answer 5

The above formula needs the 5 and the 3 switched.

Answer 6

Why is that BigFudge?

Answer 7

That's how they're solved. The upper bound is plugged in first and the lower bound is plugged in second and subtracted from the first.
http://tutorial.math.lamar.edu/Classes/CalcI/ComputingDefiniteIntegrals.aspx

Answer 8

In wikipedia it says that the simpson rule is an approximation of a definite integral and the rule is
\[\int_a^b f(x) dx \approx \frac{b-a}{6}\left[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)\right]\]
your \(f(x) = 9-x^2\), b = 5, a =3 and then you just put everything into the formula
\[\int_3^5 9-x^2 dx \approx \frac{5-3}{6}\left[(9-3^2)+4\left(9-\left(\frac{3+5}{2}\right)^2\right)+(9-5^2)\right]\]
\[ \int_3^5 9-x^2 dx \approx \frac{2}{6}\left[-28-16\right] \]
\[ \int_3^5 9-x^2 dx \approx \frac{2}{6}\left[-44\right] \]
\[ \int_3^5 9-x^2 dx \approx -\frac{44}{3} \]

Answer 9

That is how they are solved in terms of just integration. Simpson's Rule is numerical integration and the rule is:

Sn = \[\Delta x/3 (y0 + 4y1 + 2y2 +4y3 +... +2y(N-2) + 4y(N-1) + yN)\]

where \[\Delta x = (b-a)/N \]

Answer 10

Thank you plitter for straightening that out for me. It just seems weird that regular integration gives a positive answer but this one gives a negative one.

Answer 11

Regular integration still gives you the same answer just through a different process.

Answer 12

Dang, you're right. I guess I just integrated wrong in the first place. Thank you!