Find the area of the following shaded region, an unbounded region and demarcated by the equation of the curve
y = 10/(x^2 -10x +29) and the curve x=3
In the figure, the shaded area doesn't have bounds and when I plot the curves I can see that the first equation is clearly divergent towards +inf. Does it mean I can only give an approximate area?
what is the shaded region, to the left or right of x=3? I assume the region is bounded by the x-axis and one of the asymptotes. Let's take the region to the left of x=3 and to the right of x = 5 - 3√6 = -2.3485
That makes the area
A = ∫[-2.3485,3] -10/(x^2-10x-29) dx
You can do that using partial fractions, and you get
∫ 10/(x^2-10x-29) dx = 5/3∫6 log (x-(5 + 3√6))/(x-(5 - 3√6))
Hmmm. I just took a look at that integral, and it is infinite, because at either of the asymptotes you get log(0) or log(∞). It's just like trying to find the area under the curve y = 1/x^2 where the y-axis is one boundary.
I suspect you have left something out.
Yes sorry, the shaded region is to the right of x=3
yeah, well, the area is still unbounded.
You're correct. Since the first equation, y = 10/(x^2 - 10x + 29), is divergent towards positive infinity, the shaded region does not have bounds. This means that the area of the shaded region cannot be determined exactly. However, we can still approximate the area using a method called integration.
To find the approximate area, we can set up an integral. First, let's find the points of intersection between the two curves, y = 10/(x^2 - 10x + 29) and x = 3.
Setting x = 3 in the first equation:
y = 10/(3^2 - 10(3) + 29) = 10/(9 - 30 + 29) = 10/8 = 1.25
Therefore, the shaded region is bounded between x = 3 and the curve y = 10/(x^2 - 10x + 29) with y = 1.25.
Now, to approximate the area, we can set up the integral by subtracting the equation of the curve, y = 10/(x^2 - 10x + 29), from the x-axis, from x = 3 to a large positive value, let's say x = a.
The integral can be written as:
∫[3 to a] (10/(x^2 - 10x + 29)) dx
However, since the first equation is divergent towards positive infinity, we cannot determine the value of a to obtain an exact result. Therefore, we can only give an approximate area by substituting a large value for a.
For example, if we substitute a = 100, we can evaluate the definite integral numerically to get an approximate value for the area.
This approximate area represents the shaded region demarcated by the curve y = 10/(x^2 - 10x + 29) and the curve x = 3.