Area between Curves: Find Intersection & Calculate Area

In summary, to find the area of the region bounded by y=cosx, y=sin2x, 0, and x=pi/2, you must first find the intersection point between the two functions, which is pi/6. Then, you can use the formula for the area of a region, which is the integral of the top function minus the integral of the bottom function. Using this method, the area of the region is equal to 2sinx cosx. This can also be verified by using the trig identity sin 2x = 2sinx cosx.
  • #1

tony873004

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Homework Statement


Find the area of the region bounded by:
y=cosx, y=sin2x, 0, x=pi/2


Homework Equations





The Attempt at a Solution


I made a graph. I believe I'm trying to find the area I shaded.
6_1_21.GIF

red=cos(x), blue=sin(2x)

I need to find the intersection point so I will know the limits of my 2 integrals.

cosx = sin2x

But I don't know how to do this. There should be an infinate number of intersections, but I am only interested in the one that appears to happen around x=1/2 and the next one at what appears to be pi/2.

I can verify with my calculator that cos(pi/2) and sin(2pi/2) both equal 0, and that the right limit given by the problem is indeed the intersection, but that is not the case for the 1st intersection. How do I solve this? And what if the book gave the right limit as x=2. My method of eyeballing it and verifying my guess with the calculator would fail.

Assuming I find the intersection point, the next thing I was going to do is:
[tex]\left( {\int_0^{?} {\cos x} \,dx\, - \,\int_0^{?} {\sin 2x} \,dx} \right)\, + \,\,\left( {\int_{?}^{\pi /2} {\sin 2x\,dx} - \int_{?}^{\pi /2} {\cos x\,dx} } \right)[/tex]
 
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  • #2
sin(2x)=2 sinx cos x
cos x= 2sinx cos x
1=2 sin x
sin x =1/2
So the solution is arcsin (1/2). What value of sin x gives us 1/2? pi/6 of course.

Thats the systematic way of doing it, you may also have done cos x= sin (pi/2 -x) so equate them, sin (pi/2 - x)=sin 2x, 2x=pi/2 - x, 3x=pi/2, x=pi/6.

The 2nd way is only useful for angles less than pi/2, which means it can be used here.

Now that you have both intersection points just integrate as how you were going to do next, knowing the ?'s is pi/6.
 
  • #3
Thanks for your reply.

sin(2x)=2 sinx cosx
I guess that's just a trig identity I forgot? Playing with it on my calculator, I can see that it works for all values of x.

cosx = 2 sinx cosx
Playing with this on my calculator, it does not work for all values of x. Is this a trig identity, or specific to this problem?

If specific to this problem, from sin(2x)=2 sinx cosx i get
cosx = sin(2x)/2sin x

How did you get cosx = 2 sinx cosx ?
 
  • #4
Sin 2x = 2sinx cos x is a identity you just forgot.

Cos x= 2sinx cos x is specific to the problem. When we want to find points of intersection, we set the 2 functions to be equal, then solve. The functions we have are sin 2x and cos x. Since sin 2x=2sinx cos x for all x, we sub that in, then we have to solve cos x=2sinx cos x.

I don't know why you got cos x= sin(2x)/2 sin x, that's not needed.

cos x = 2sin x cos x

Divide both sides by 2 cos x, sin x=1/2. Solving for that, arcsin 1/2=pi/6
 
  • #5
The Indentity can be easily seen from expanding the formula for sin(x+y), letting y=x.
 

1. What is the formula for finding the area between two curves?

The formula for finding the area between two curves is ∫(upper curve - lower curve) dx. This means taking the integral of the difference between the two curves with respect to the x-axis.

2. How do you find the intersection points of two curves?

To find the intersection points of two curves, set the equations equal to each other and solve for the values of x. These values will be the x-coordinates of the intersection points. To find the y-coordinates, plug the x-values back into one of the original equations.

3. Can you find the area between curves if they do not intersect?

Yes, it is still possible to find the area between two curves even if they do not intersect. In this case, you would need to break the region into smaller sections where the curves do intersect and find the area for each section separately, then add them together.

4. How do you handle negative areas when finding the area between curves?

If the area between two curves results in a negative value, this means that the lower curve is above the upper curve in certain regions. To find the actual area, you would need to take the absolute value of the integral to make it positive.

5. Can you use the same formula for finding the area between three or more curves?

Yes, the same formula can be used for finding the area between three or more curves. You would just need to adjust the limits of integration to include all of the curves and subtract the lower curves from the upper curves.

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