one expression is y=log(-100493/40000)x the x is next to the (-100493/40000)
same with the other expression
y=atan(-100493/40000)x,
y=exp(100493/40000x)
approximate the region bounded by the given curves. be sure to specify a range of x and y that results in a good picture of the region. you need two integral to represent the area
It is still not clear to me what the functions should be.
I have put the variable x inside of all the functions (log, atan and exp) for the simple reason that log() does not take a negative argument.
Check the following link and confirm that the functions are correct:
http://img156.imageshack.us/i/1299606125.png/
You can create the same plot using:
wxplot2d([log((-100493/40000)*x),atan((-100493/40000)*x),exp((100493/40000)*x)], [x,-2,0],[y,-0,2])$
If the functions are correct, you will note that the closed region bounded by the three functions has three intersections, P1, P2 and P3.
P1 (Red/Blue) at x=-1.5 approx.
P2 (Blue/Green) at x=-0.5 approx.
P3 (Green/Red) at x=-0.25 approx.
You will need to compute the coordinates of the three intersections (x1,y1),...(x3,y3)and do two integrations to find the area, I1 between x1 and x2, and I2 between x2 and x3.
Post if you have other questions.
so what will the arEA BE?
Do you have access to the software?
If you do, you could do the calculations, and I can verify them for you.
In any case, first find the intersections which will give the limits of integration. Do the integrations using the menu if you wish.
If you have problems coding the statements, post where you have difficulties.
By the way, have you verified the plots of the function and the region? Are they as you expected? There is no point working on the integrals if the curves are not confirmed.
To find the limits of integration, i.e. the intersections of the functions, you
define first the functions:
f1(x):=log(((-100493)*x)/40000) [blue]
f2(x):=atan(((-100493)*x)/40000) [red]
f3(x):=exp((100493*x)/40000) [green]
and use
find_root(f1(x)-f2(x), x, -1.7, -1.3);
to get x=-1.469788218266142
You can find the other intersections in a similar way.
I get 0.376 for the total area.
To approximate the region bounded by the given curves, we first need to find the points of intersection between the curves.
Starting with the expressions:
1. y = log(-100493/40000)x
2. y = atan(-100493/40000)x
3. y = exp(100493/40000x)
To find the intersection points between the first and second curves, we set them equal to each other and solve for x:
log(-100493/40000)x = atan(-100493/40000)x
Next, we take the exponential function of both sides to eliminate the logarithm:
e^log(-100493/40000)x = e^atan(-100493/40000)x
Simplifying, we get:
(-100493/40000)x = atan(-100493/40000)x
Now, to find the intersection points between the second and third curves, we set them equal to each other and solve for x:
atan(-100493/40000)x = exp(100493/40000x)
Since the trigonometric and exponential functions do not have direct algebraic solutions, we need to use numerical methods or graphing software to find the approximate intersection points.
Once we have determined the intersection points, to find the range of x and y that results in a good picture of the region, we look for the x-values that bound the region and determine the corresponding y-values based on the given expressions.
Once we have identified the range of x and y values, we can proceed to calculate the integrals to represent the area between the curves. Since you mentioned that two integrals are needed, we might need to divide the region into two separate areas if the curves intersect multiple times.
To calculate the area between the curves, set up integrals based on the x-values that bound the region and the corresponding y-values from the expressions. Evaluate the integrals to find the approximate area of the region between the curves.
Note: Since we don't have the exact intersection points or the specific range of x-values, I am unable to provide the final numerical calculations for the areas.