Estimate the Value

Question

given that 7≤f(x)≤9 for 2≤x≤3, estimate the value of ∫3(top) 2(bottom) f(x) dx

________________ ≤ ∫3(top) 2(bottom) f(x) dx ≤ _____________

I don't understand what this is asking. If you could help me that would be much appreciated!

Answer 1

It is asking to estimate the definite integral of f(x) over the interval

2 <= x <= 3.

In other words, the area under the curve of f(x) over that interval.

In general, if f(x) <= k over an interval of width w then

the area under the curve cannot exceed kw.

Similarly for the lower bound.

Your interval has width 1.

Therefore your integral must be between 7 and 9.

Answer 2

Trapezoidal rule says:

(1) l the base area is 7*(3-2) = 7

(2) the right triangle formed that supplants the function

has area (1/2)*1*2 = 1

so the trapezoidal estimate is 8

in other words, we are assuming the function is below 8 half of the time

and greater than 8 the other half, as it is (assumed) a continuous function

between 7 and 9 on the interval