The greatest value of Riemann sum and integral of a function on the interval [0,10]

Question

My answer was D, but the correct answer is B, I do not no why my answer is wrong and why B is the correct answer and why other answers are wrong. Could anyone explain this for me?

Answer 1

hint

$f $ is decreasing at $\Bbb R $, thus

For $1\le j\le 10$

$$f (x_j)\le f (\frac {x_{j-1}+x_j}{2})\le f (x_{j-1}) $$ and $$\forall x\in [x_{j-1},x_j] \;\; f (x)\le f (x_{j-1}) $$

hence, the answer is $B $.

Answer 2

The function $f(x) = e^{-x}$ is strictly decreasing and hence the left-most point in the interval will maximize the value of the Riemann Sum. To understand why you answer is wrong remember that the integral is actual area under the curve. It's always no bigger than any upper sum and no less than any lower sum.