Question

Function f(x) is positive, increasing and concave up on the closed interval [a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the upper sum, lower sum, and trapezoidal rule approximations for the value of the integral from a to b of f of x dx. Which one of the following statements is true?

Answer 1

https://i.gyazo.com/c8e5790755c07810fed45ac088898c32.png

Answer 2

@jim_thompson5910 :)

Answer 3

A?

Answer 4

@zepdrix

Answer 5

Your answer is correct. Knowing you're right is less than half the battle, though. Do you know why you're right?

Answer 6

I justified it by saying trapezoidal is area under curve while lower sum is well lower sum and upper sum is upper sum. Yeah it was kind of a guess, explanation would be appreciated @SithsAndGiggles

Answer 7

First some comment:

(1) The trapezoidal sum is only exactly equal to the area under the curve in the case of linear functions.

(2) "Lower" and "upper" in this context don't mean that the value of the lower/upper sums are the smallest/largest of the two (respectively). "Lower" instead refers to using the lower limit of each subinterval of the partition to determine the height of the approximating rectangle, while "upper" uses the upper limit of each subinterval. In this sketch, I've drawn an arbitrary positive function with just one subinterval that covers the interval \([a,b]\), which means \(a\) is the lower limit and \(b\) is the upper limit. The lower sum approximation to the area under the curve over \([a,b]\) is given by a rectangle whose height is determined by the value of the function at \(a\), so
\[\int_a^bf(x)\,\mathrm{d}x\approx\text{Lower sum}=f(a)(b-a)\]Meanwhile, the upper sum uses a rectangle whose height is determined by \(b\), so
\[\int_a^bf(x)\,\mathrm{d}x\approx\text{Upper sum}=f(b)(b-a)\]