# Help find error bound of trapezoidal quadrature

I'm having trouble finding the error bound of this function. My professor says it's "trivial" and thus, he refused to offer me any help beyond a simple hint I kind of knew anyway.

I am given this integral and I need to find it's error when we calculate it using the trapezoidal rule.

$$\int_{0.75}^{1.3}((sinx)^2 - 2xsinx + 1)dx$$

So I know

$$\epsilon = |\frac{h^3}{12}f''(\xi)|$$ where $\xi \in (0.75,1.3)$

Now, I find the second derivative:

$$f'(x) = 2sinxcosx - 2xcosx - 2sinx$$ $$f''(x) = -8sinxcosx + 6sinx + 2xcosx$$

Now, the only hint my professor did give me was that since we can't get the maximum of $f''(\xi)$ analytically we can take advantage of the property that both $sinx$ and $cosx$ are bounded by 1 from above. This means the worst that any of these trig functions could possibly be is 1. The issue is that $sinx$ and $cosx$ are 1 at different values.

Can anyone give me any hints on what I need to do to make the leap to understanding what is going on? I feel really lost.

EDIT: What I truly need help on is understanding how to maximize $f''(\xi)$

What I have done is (given $h=0.55)$:

$$\epsilon = |\frac{0.55}{12} * -8sin(\xi)cos\xi + 6sin\xi + 2(\xi)cos\xi|$$

and knowing that on $(0.75,1.3)$ $f''(\xi)$ is maximized at 1.3 just plug that in

$$\epsilon = |\frac{0.55}{12} * -1.3sin(1.3)cos1.3 + 6sin1.3 + 2(1.3)cos1.3| = 0.0612099$$

but the book says

$$\epsilon = 0.02444080544$$

and I have no idea how they got this.

Thanks!

• You want the modulus and |a+b|<=|a|+|b| – Paul Nov 22 '14 at 9:05
• @Paul if you're talking about using the triangle inequality to split $\frac{h^3}{12}f''(\xi)$ I know that. – John Nov 22 '14 at 17:49
• @Paul I have posted my work – John Nov 22 '14 at 18:03

• Thank you for your answer, however the book says the error bound should be 0.02444080544. Plugging in $h=0.55$ you get 0.39416 using your solution, which is rather large. How did you get the numbers you did? – John Nov 22 '14 at 17:53