Question

hi..I'm ronsthel gordula. Bs. math student .my question is about trapezoidal rule and its truncation error.

trapezoidal rule is a technique of approximating definite integrals with the use of trapezoids and calculating its area..

if the equation like this..integral of (9-x^2)dx from 1to 3..
what is the value of truncation error..?

Answer 1

how many trapezoids are you using to approximate?

Answer 2

if n=4
can you solve this equation for me and find the truncation error

\[int_{1}^{3}(9-^{2})dx\]

Answer 3

f[x_] = 9 - x^2
Integrate[f[x], {x, 1, 3}]=9.33333
2/(2*4)*(f[1] + 2*f[1.5] + 2*f[2] + 2*f[2.5] + f[3])=9.25

error = 9.3333-9.25 = 1/12

Answer 4

can you give another example..?

Answer 5

http://www.dummies.com/how-to/content/how-to-approximate-area-with-the-trapezoid-rule.html

Answer 6

do you have another information about trapezoidal rule..?..and why the simson's rule is the fast estimated to find area under the curve yhan trapezoidal rule..?

Answer 7

With the trapezoid rule, instead of approximating area by using rectangles (for example, as you do with the left and right sum methods), you approximate area with — can you guess? — trapezoids.

Because of the way trapezoids hug the curve, they give you a much better area estimate than either left or right rectangles. And it turns out that a trapezoid approximation is the average of the left rectangle and right rectangle approximations. Can you see why? (Hint: The area of a trapezoid is the average of the areas of the two corresponding rectangles in the left and right sums.)With the trapezoid rule, instead of approximating area by using rectangles (for example, as you do with the left and right sum methods), you approximate area with — can you guess? — trapezoids.

Because of the way trapezoids hug the curve, they give you a much better area estimate than either left or right rectangles. And it turns out that a trapezoid approximation is the average of the left rectangle and right rectangle approximations. Can you see why? (Hint: The area of a trapezoid is the average of the areas of the two corresponding rectangles in the left and right sums.)

Answer 8

trapezoidal rules uses straight lines to approximate the arc length



while simpson's rule uses parabolic arcs to approximate the area.

so the error will be smaller with simpson's rule... as you are using a arc rather than a straight line.

Answer 9

here are some good notes on approximating integrals

http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx