Find the area of the shaded region bounded by y=1/x^2+x , x = 2, x = 4, and the x-axis.

Answer 1

To find the area of the shaded region, we need to integrate the function y = 1/x^2 + x over the interval from x = 2 to x = 4.

The integral that represents the area is given by:

∫[2, 4] (1/x^2 + x) dx

We can split the integral into two separate integrals to make the computation easier:

∫[2, 4] (1/x^2) dx + ∫[2, 4] x dx

For the first integral, we have:

∫[2, 4] (1/x^2) dx = [-1/x] [2, 4] = (-1/4) - (-1/2) = 1/2 - 1/4 = 1/4

For the second integral, we have:

∫[2, 4] x dx = [(x^2)/2] [2, 4] = (4/2) - (2/2) = 2 - 1 = 1

Therefore, the area of the shaded region is given by:

1/4 + 1 = 5/4 (or 1.25)

So, the area of the shaded region is 5/4 or 1.25 square units.