Simpson's Rule/Trapezoidal Approximation - Error rate help

[Asphyxiated]

Homework Statement

\int^{ \pi}_{0} sin(x)dx \;\;\;\;\;\;\;\; dx=\frac{ \pi}{2}

Homework Equations

Trapezoidal Approximation:

|f''(x)| \leq M \;\;\;\;\; for \;\;\;\;\; a \leq x \leq b

\frac {b-a}{12}(M)(dx)^{2} = Error

Simpson's Rule:

|f^{(4)}(x)| \leq M \;\;\;\;\; for \;\;\;\;\; a \leq x \leq b

\frac{b-a}{180}(M)(dx)^{4} = Error

The Attempt at a Solution

Ok so I have found the correct estimations using both methods easily, the trapezoidal approximation is: 1.5708 and Simpson's Rule is: 2.0944, those numbers check out in the back of the book, but when it comes to finding the error I think that it should be 0 because the max (M) is zero for both the second and fourth derivative but the book says otherwise. Heres what I did:

y=sin(x)

y'=cos(x)

y''=-sin(x)

y^{(3)}= -cos(x)

y^{(4)}= sin(x)

and Trapezoidal Rule using y'' is:

|y''( \pi)|=0

and

|y''(0)|=0

and that follows the same for y^{(4)} so M is 0 and thus the entire equation is 0 and Error = 0 but the book states that the error for the trapezoidal approximation is:

\frac { \pi^{3}}{48} \;\;\;\;\; or \;\;\;\; .65

and the Error for Simpson's Rule is:

\frac { \pi^{5}}{2880} \;\;\;\; or \;\;\;\; .1

I don't see how they got this... but I don't think the actual error rate is zero either because if it were then the trapezoidal and simpson approximation would be exactly equal, so where did I go wrong?

 

[Mark44]

Homework Statement

\int^{ \pi}_{0} sin(x)dx \;\;\;\;\;\;\;\; dx=\frac{ \pi}{2}

Homework Equations

Trapezoidal Approximation:

|f''(x)| \leq M \;\;\;\;\; for \;\;\;\;\; a \leq x \leq b

\frac {b-a}{12}(M)(dx)^{2} = Error

Simpson's Rule:

|f^{(4)}(x)| \leq M \;\;\;\;\; for \;\;\;\;\; a \leq x \leq b

\frac{b-a}{180}(M)(dx)^{4} = Error

The Attempt at a Solution

Ok so I have found the correct estimations using both methods easily, the trapezoidal approximation is: 1.5708 and Simpson's Rule is: 2.0944, those numbers check out in the back of the book, but when it comes to finding the error I think that it should be 0 because the max (M) is zero for both the second and fourth derivative but the book says otherwise.

f''(x) = -sin(x) and f⁽⁴⁾(x) = sin(x). The maximum value of the absolute values of these functions is not zero. What you're looking for is the maximum value over the entire interval [0, pi].

Heres what I did:

y=sin(x)

y'=cos(x)

y''=-sin(x)

y^{(3)}= -cos(x)

y^{(4)}= sin(x)

and Trapezoidal Rule using y'' is:

|y''( \pi)|=0

and

|y''(0)|=0

and that follows the same for y^{(4)} so M is 0 and thus the entire equation is 0 and Error = 0 but the book states that the error for the trapezoidal approximation is:

\frac { \pi^{3}}{48} \;\;\;\;\; or \;\;\;\; .65

and the Error for Simpson's Rule is:

\frac { \pi^{5}}{2880} \;\;\;\; or \;\;\;\; .1

I don't see how they got this... but I don't think the actual error rate is zero either because if it were then the trapezoidal and simpson approximation would be exactly equal, so where did I go wrong?