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Split up the interval [2, 5] into 
equally spaced subintervals, then consider the value of 
at the right endpoint of each subinterval.
The length of the interval is
, so the length of each subinterval would be 
. This means the first rectangle's height would be taken to be 
when 
, so that the height is 
, and its base would have length 
. So the area under over the first subinterval is 
.
Continuing in this fashion, the area under over the 
th subinterval is approximated by 
, and so the Riemann approximation to the definite integral is
and its value is given exactly by taking
. So the answer is D (and the value of the integral is exactly 39).
The length of the interval is
Continuing in this fashion, the area under
and its value is given exactly by taking
y² - 4y - 2x - 4 = 0
<u> +2x +4 </u> <u>+2x + 4 </u>
y² - 4y = 2x + 4
<u> +4 </u> <u> +4 </u>
(y - 2)² = 2x + 8
<u> -8 </u> <u> -8 </u>
(y - 2)² - 8 = 2x
(y - 2)² - 4 = x
x = (y - 2)² - 4
<em>Note: this is a parabola whose axis of symmetry is y = 2 and vertex is (-4, 2)</em>