To find the area of the shaded region, we need to integrate the function y=1/x^2+x between x=2 and x=4.
The integral can be set up as follows:
∫[2, 4] (1/x^2 + x) dx
To integrate 1/x^2 + x, we can split it into two separate integrals:
∫[2, 4] 1/x^2 dx + ∫[2, 4] x dx
The first integral is the integral of 1/x^2, which can be computed as -1/x: (-1/x) evaluated from 2 to 4.
= [(-1/4) - (-1/2)]
= (1/2 - 1/4)
= 1/4
The second integral is the integral of x, which can be computed as (x^2/2): (x^2/2) evaluated from 2 to 4.
= [(4^2/2) - (2^2/2)]
= (16/2 - 4/2)
= (8 - 2)
= 6
Now we need to add the two integrals together:
(1/4) + 6
= 25/4
Therefore, the area of the shaded region is 25/4.
Find the area of the shaded region bounded by y=1/x2+x , x = 2, x = 4, and the x-axis. (1 point) Responses 25/4 25 4 29/8 29 8 21/2 21 2 43/4 43 4 101/8
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1 answer