Question

Function f(x) is positive, decreasing 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 left sum, right sum, and trapezoidal rule approximations for the value of integral from a to b f(x)dx. Which one of the following statements is true?
a) Left sum < trapezoidal rule value < Right sumb) Left sum < Right sum < trapezoidal rule valuec) Right sum

Answer 1

Answer:

Right sum < trapezoidal rule value < Left sum

Non of the completed options are valid. Option c is incomplete but it starts correctly. If there are no more options, the answer is (c).

Step-by-step explanation:

Lets call a = x0, x1, x2, x3 and x4 = b the endpoints of the intervals in increasing order. Lets also call L the length of each subinterval. Note that b-a = 4L. Lets denote yi = f(xi) for i in {0,1,2,3,4}.

Lets compute each sum:

Left sum = x0*L + x1*L + x2*L + x3*L = (y0+y1+y2+y3)L

Similarly

Right sum = (y1+y2+y3+y4)L

and trapezoidal rule value = L((y0)/2 + y1+ y2 + y3+ (y4)/2)

Since f is decreasing, we have that y0 > y1 > y2 > y3 > y4. Therefore, y0 > y4 and, as a result, Left sum > Right sum, because the Right sum has y4 instead of y0 multiplying by L.

On the other hand, multiplying L we have (y0)/2 + y1+ y2 + y3+ (y4)/2 on the trapezoidal rule value, thus it has half of y0 and half of y4, and as a consecuence, the trapezoidal sum is between the left sum and the right sum.

The correct answer is

Right sum < trapezoidal rule value < Left sum