Find the area of the shaded region
y= 2x/(x^2+4)
from x= -1 and x= 1
Note: People have been telling me that the two lines do not intersect but they do there has to be an answer. please help me.
Thanks
Here is what your curve looks like from x = -1 to +1
http://www.wolframalpha.com/input/?i=y%3D+2x%2F%28x%5E2%2B4%29+from+-1+to+1
So where is the shaded region?
Are you finding the area between the curve and the x-axis?
(That is the usual case for this question, but let me know before I do it)
Yes, I have to find the area between the curve and the x-axis. Thanks for the help.
To find the area of the shaded region between the graph of the function y = 2x/(x^2+4) and the x-axis, we need to integrate the absolute value of the function over the interval from x = -1 to x = 1.
Step 1: Determine the equation of the function.
The given equation is y = 2x/(x^2+4).
Step 2: Verify if the two lines intersect.
To determine if the lines intersect, we need to find the x-values where the function y = 0, since the shaded region is bounded by the x-axis. Set the function equal to zero:
0 = 2x/(x^2+4).
To solve this equation, multiply both sides by (x^2+4):
0 = 2x.
Since this equation simplifies to 0 = 0, it means that the line y = 0 intersects the function at all points when the function is defined.
Therefore, the two lines do intersect at x = -1, x = 0, and x = 1, and we can proceed with finding the area of the shaded region.
Step 3: Integrate the absolute value of the function between the given x-values.
To find the area between the function and the x-axis, we need to integrate the absolute value of the function over the interval from x = -1 to x = 1. The absolute value ensures that the area is positive.
∫[a,b] |f(x)| dx, where a = -1 and b = 1:
∫[-1,1] |2x/(x^2+4)| dx.
Step 4: Simplify the absolute value expression.
Since the absolute value function creates a piecewise definition, we need to consider two separate intervals separated by the x-values where the function changes sign (x = -1, x = 0, and x = 1).
∫[-1,1] |2x/(x^2+4)| dx = ∫[-1,0] -(2x/(x^2+4)) dx + ∫[0,1] (2x/(x^2+4)) dx.
Step 5: Evaluate the integrals.
For the first interval [-1,0], we integrate -(2x/(x^2+4)) dx:
∫[-1,0] -(2x/(x^2+4)) dx = -ln|x^2+4| |_1^0 ≈ -ln(4).
For the second interval [0,1], we integrate (2x/(x^2+4)) dx:
∫[0,1] (2x/(x^2+4)) dx = ln|x^2+4| |_0^1 ≈ ln(5).
Step 6: Find the difference between the integrals to obtain the area of the shaded region.
To find the area of the shaded region, we calculate the difference between the two integrals:
Area = (∫[-1,0] -(2x/(x^2+4)) dx) - (∫[0,1] (2x/(x^2+4)) dx)
= -ln(4) - ln(5)
= ln(1/4) - ln(5)
≈ -1.386 - 1.609
≈ -2.995.
Note: The negative sign indicates that the area is below the x-axis, which means the shaded region is considered with a negative area.