Let $P= 0, \frac{1}{2},1,2$
Find the upper, lower, and exact area and whether the lower or upper sum is more accurate
Starting with the left:
LH: $$\sin(0)(\frac{1}{2}-0)+\sin(\frac{.5}{5})(1-\frac{1}{2})+\sin(\frac{1}{5})(2-1)$$
$$0+.0499+.19=.248$$
RH:
$$\sin(\frac{.5}{5})(\frac{1}{2}-0)+\sin(\frac{1}{5})(1-\frac{1}{2})+\sin(\frac{2}{5})(2-1)$$
$$=.0499+.19+.389=.61$$
Neither of these is close to the book which says that the sum approximately equal to $.5386$ is closest. What is wrong with my calculation?
It looks like you forgot $\color{red}{\left(1-\dfrac12\right)}$:
$$\sin\left(\frac{.5}{5}\right)\left(\frac{1}{2}-0\right)+\sin\left(\frac{1}{5}\right)\color{red}{\left(1-\frac{1}{2}\right)}+\sin\left(\frac{2}{5}\right)(2-1)$$
$$\approx.0499+.1987\times\color{red}{\frac12}+.3894$$
$$\approx.5386$$