Find the area of the shaded region
y= 2x/(x^2+4)
from x= -1 and x= 1
To find the area of the shaded region, we first need to determine the boundaries of the region. In this case, the boundaries are given by x = -1 and x = 1.
To find the area of the shaded region between two curves, we need to subtract the area under one curve from the area under the other curve.
Let's denote the shaded region as Area_s and the curve y = 2x/(x^2 + 4) as Curve_1.
Now, let's find the area under Curve_1 between x = -1 and x = 1. We'll call this Area_1.
To calculate Area_1, we can integrate the function y = 2x/(x^2 + 4) with respect to x, over the given bounds.
∫[x=-1 to 1] 2x/(x^2 + 4) dx
To integrate this expression, we can use a substitution u = x^2 + 4. The derivative du with respect to x is 2x dx.
Now, let's calculate du/dx = 2x:
du = 2x dx
Substituting the value of du in the integral, we get:
∫[u=3 to 5] (1/u) du
Evaluating this integral, we get:
ln|u| |[u=3 to 5]
Substituting back the value of u = x^2 + 4, we get:
ln|(x^2 + 4)| |[x=-1 to 1]
Now, let's find the area under the x-axis, which is the area under the curve y = 0 between the same bounds x = -1 and x = 1. We'll call this Area_2.
The area under the x-axis is given by:
∫[x=-1 to 1] 0 dx
Since the integral of a constant is equal to the constant times the interval length, we can simply calculate:
0 * (1 - (-1)) = 0
Finally, we can find the area of the shaded region, Area_s, by subtracting Area_2 from Area_1:
Area_s = Area_1 - Area_2
= ln|(x^2 + 4)| |[x=-1 to 1] - 0
Evaluating this expression, we get the final result for the area of the shaded region.