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

Answer 1

To find the area of the shaded region, we need to integrate the function y=1/x^2+x from x=2 to x=4, then take the absolute value of the result.

∫[2, 4] (1/x^2+x) dx

To integrate this function, we can split it up into two separate integrals:

∫[2, 4] (1/x^2) dx + ∫[2, 4] x dx

To integrate 1/x^2, we can use the power rule for integration:

∫(1/x^2) dx = -1/x

To integrate x, we can use the power rule for integration:

∫x dx = (1/2)x^2

Now, we can evaluate the definite integrals:

= [-1/x] from 2 to 4 + [(1/2)x^2] from 2 to 4
= [-1/4 - (-1/2)] + [(1/2)(4^2) - (1/2)(2^2)]
= [-1/4 + 1/2] + [8/2 - 2/2]
= [1/4] + [6/2]
= 1/4 + 3
= 13/4

Therefore, the area of the shaded region is 13/4 square units.