How do you find the area using the trapezoidal approximation method, given (x^2-x)dx, on the interval [0,2] with n=4?

Aug 18, 2017

The interval $\left[0 , 2\right]$ is split into $4$ equal intervals of $1 / 2$. Pictured are the four functions, the intervals, and the areas created by making trapezoids from the function points.

The first three intervals are actually triangles since they have $0$-points that rest on the $x$-axis.

Using the points

• $f \left(0\right) = 0$
• $f \left(\frac{1}{2}\right) = - \frac{1}{4}$
• $f \left(1\right) = 0$
• $f \left(\frac{3}{2}\right) = \frac{3}{4}$
• $f \left(2\right) = 2$

we can calculate all the needed areas. The area for a trapezoid is given by $A = \frac{1}{2} h \left({b}_{1} + {b}_{2}\right)$. Areas below the $x$-axis are negative.

${A}_{\text{total}} = - \frac{1}{2} \left(\frac{1}{2}\right) \left(\frac{1}{4}\right) - \frac{1}{2} \left(\frac{1}{2}\right) \left(\frac{1}{4}\right) + \frac{1}{2} \left(\frac{1}{2}\right) \left(\frac{3}{4}\right) + \frac{1}{2} \left(\frac{1}{2}\right) \left(\frac{3}{4} + 2\right)$

$= \frac{3}{4}$