How do you approximate of int sinx(dx) from [0,pi] by the trapezoidal approximation using n=10?

Oct 15, 2015

See the explanation section.

Explanation:

To approximate the Integral ${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ using trapezoidal approximation with $n$ intervals.

In this question we have:
$f \left(x\right) = \sin x$
$\left\{a , b\right] = \left[0 , \pi\right]$, and
$n = 10$.

So we get
$\Delta x = \frac{b - a}{n} = \frac{\pi - 0}{10} = \frac{\pi}{10}$

The endpoints of the subintervals are found by beginning at $a = 0$ and successively adding $\Delta x = \frac{\pi}{10}$ to find the points until we get to ${x}_{n} = b = \pi$.

${x}_{0} = 0$, ${x}_{1} = \frac{\pi}{10}$, ${x}_{2} = \frac{2 \pi}{10}$, ${x}_{3} = \frac{3 \pi}{10}$ . . . ${x}_{9} = \frac{9 \pi}{10}$, and ${x}_{10} = \frac{10 \pi}{10} = 10 = b$

Now apply the formula (do the arithmetic) for $f \left(x\right) = \sin x$.

${T}_{4} = \frac{\Delta x}{2} \left[f \left({x}_{0}\right) + 2 f \left({x}_{1}\right) + 2 f \left({x}_{2}\right) + \cdot \cdot \cdot 2 f \left({x}_{9}\right) + f \left({x}_{10}\right)\right]$