Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.

Question

f(x) = 4x x2 + 5 , 1 ≤ x ≤ 3

Answer 1

f(x) = 4x/(x^2 + 5)

Let

U = x^2 + 5

Then dU = 2x

so 2 dU = 4x

integral [ f(x) ] = 2 * integral [ dU/U]

= 2 * [ ln |U| ]

= 2 * ln | x^2 + 5|

limit as x-->3 : 2 * ln |14|

limit as x-->1 : 2 * ln |6|

Subtracting these limits:

2 [ ln 14 - ln 6] = 2 ln (14/6) = 2 ln (7/3)

The approximation is

1.6945957207744072274202150130413

Answer 2

Divide the interval [1,3] into n equal subintervals each of length Δx = (3-1)/n = 2/n

Let x_, = right endpoint of interval i = 1 + 2i/n

Then, the integral (area, since f(x)) = 4x² + 5 is nonnegative on [1,3]) is:

lim_n→∞ [∑_{(from i = 1 to n)} (f(x_i)Δx)] = lim_n→∞[(2/n)∑_{(from i = 1 to n)}(4(x_i)² + 5)]

You could also use left endpoints, midpoints, or any point in each interval.