Not sure about this question..

Question

The rectangles in the graph below illustrate a left endpoint Riemann sum for f(x)=x^2/9 on the interval [3,7]. The value of this left endpoint Riemann sum is _________ , and this Riemann sum is ________. the area of the region enclosed by y = f(x), the x-axis, and the vertical lines x = 3 and x = 7.

Answer 1

interval x=3 to x=7

area for f(x) = x^2/9

= integral of x^2/9 = x^3/27 + a constant that you can ignore as it cancels out

evaluated from 3 to 7

= 7^3/27 - 3^3/27 = 343/27 -1 = 316/27

= about 11.7 square units

do the Reiman sums with left end points and you get a little less

right end points and a little more

take their average and you get like Goldie Locks in the 3 bears bedtime story

poridge too hot, too cold, then just about right

or check with geometry

it's sort of a "flat" parabola

so construct a rectangle which is (7-3)=4 by (1/9)(7^2-3^2) = 4(40/9) = 160/9 = 17 or 18

which is larger than 11.7 but about in the right range, as the parabola area < the rectangle