Find the Upper and Lower Bounds based on sums of rectangles

Find the Upper and Lower Bounds based on sums of rectangles for the following integrals. integrals^1_0 (x^2+4)^-1 dx. Use the partition P = {0, 1/2, 1}. Integral^2_1 x^-3 dx. Use the partition P = {1, 3/2, 2}.

Solution

a)

Values :   f(0) = 1/4     f(1/2) = 4/17    f(1) = 1/5

Now, this is decreasing graph, as x increases.

First for lower bound:

integration = (1/2- 0 )*f(1/2) + (1-1/2)*f(1)     //i.e. we put the smallest of f(x) for height to get lower bound

i.e. lower bound is: 37/190

For upper bound:

= (1/2-0)*f(0) + (1-1/2)*f(1/2)

= 31/76

b)

Values:      f(1) = 1          f(3/2) = 8/27                  f(2) =1/8

lower bound is:   (1/2)*f(3/2) + (1/2)*f(2) = 91/(27*16)

upper bound is:   (1/2)*f(1) + (1/2)*f(3/2) = 35/54