Finding Area Between Cosine Curves [0, pi] Using Integration

In summary, when finding the area between the curves y=4cosx and y = 4cos(2x) on the interval [0, pi], it is necessary to use a trigonometric identity to solve for the points of intersection. One useful identity is cos(2x) = cos(x), which can be rewritten in several other forms.
  • #1
char808
27
0

Homework Statement



Find the area between the curves y=4cosx and y = 4cos(2x) [0, pi]

Homework Equations





The Attempt at a Solution



I know I need to integrate this, but I get hung up finding the intersection of the two lines so I can split it into two different areas.

4cosx = 4 cos(2x)

0 = cos(2x) - cos(x) I think I need to use a trig identity here, but I'm not sure.
0 = sin(2x)sin(x)

x= pi?
 
Physics news on Phys.org
  • #2
char808 said:

Homework Statement



Find the area between the curves y=4cosx and y = 4cos(2x) [0, pi]

Homework Equations





The Attempt at a Solution



I know I need to integrate this, but I get hung up finding the intersection of the two lines so I can split it into two different areas.

4cosx = 4 cos(2x)

0 = cos(2x) - cos(x) I think I need to use a trig identity here, but I'm not sure.
Finding the points of intersection would be a very good idea, and a trig identity would be very useful.
char808 said:
0 = sin(2x)sin(x)
?? How did you go from cos(2x) - cos(x) to sin(2x)sin(x)?
char808 said:
x= pi?
That looks like a guess.
 
  • #3
Yes, I'm at a loss about where to go from here:

0 = cos(2x) - cos(x)

Possibility: (I am admittedly weak when it comes to trig, I could/can figure this out with almost any other function.)

0=cos(2x-x)
0= cos(x)

x= pi/2?
 
  • #4
char808 said:
0 = cos(2x) - cos(x) I think I need to use a trig identity here, but I'm not sure.

Yes indeed you do. What identities do you know for cos(2x)?
 
  • #5
I don't. If you have a reference link I can go study I would appreciate that. Not covered in the book I have.
 
  • #6
char808 said:
Yes, I'm at a loss about where to go from here:

0 = cos(2x) - cos(x)

Possibility: (I am admittedly weak when it comes to trig, I could/can figure this out with almost any other function.)

0=cos(2x-x)
cos(2x) - cos(x) [itex]\neq[/itex] cos(2x - x)
char808 said:
0= cos(x)

x= pi/2


?
 
  • #7
char808 said:
I don't. If you have a reference link I can go study I would appreciate that. Not covered in the book I have.

Did you study trig at any time? If so, what did you do with your book?

khanacademy.org has a lot of lectures about a variety of math stuff. You might start there.
 
  • #8
Guessing also works. For example, cos(2x)=cos(x) means that, at one point, 2x=x (there are also other points, however). Only one point can satisfy 2x=x.
 
  • #9
Mark44 said:
Did you study trig at any time? If so, what did you do with your book?

khanacademy.org has a lot of lectures about a variety of math stuff. You might start there.

I last studied trig ~7-8 years ago. I probably sold the book.

I will check out the link.
 
  • #10
hi char808 ! :smile:

you need to learn the standard trigonometric identities …

cosA - cosB = 2 sin((A+B)/2) sin((A-B)/2) would help :wink:
 
  • #11
One that I found helpful was
cos(2x) = cos2(x) - sin2(x)

Two other forms of this are
cos(2x) = 2cos2(x) - 1
cos(2x) = 1 - 2sin2(x)

One of these can be used to write your equation cos(2x) = cos(x) as a quadratic in form.
 

FAQ: Finding Area Between Cosine Curves [0, pi] Using Integration

1. What is integration and why is it important in science?

Integration is the process of bringing together different elements or parts to form a whole. In science, it is important because it allows us to understand complex systems and how various components interact with each other. It also helps us make connections between different disciplines and expand our understanding of the world.

2. How do scientists integrate different theories or models?

Scientists use a variety of methods to integrate different theories or models. One common approach is through mathematical modeling, where they use equations and data to create a unified model. They also use experiments and observations to test and refine their integrated theories.

3. Can integration be applied to all fields of science?

Yes, integration can be applied to all fields of science. It is a fundamental aspect of the scientific method and is used in fields such as biology, chemistry, physics, and earth sciences.

4. What are some challenges in integrating different scientific disciplines?

Some challenges in integrating different scientific disciplines include differences in terminology, methodologies, and underlying assumptions. It can also be difficult to find a common language or framework to integrate different theories and models.

5. How does integration contribute to scientific progress?

Integration plays a crucial role in scientific progress by allowing scientists to build upon existing knowledge and make connections between different fields. It also helps identify gaps in our understanding and leads to new discoveries and advancements. Additionally, integration can lead to more comprehensive and accurate models, improving our overall understanding of the natural world.

Similar threads

Calculating Area Under a Curve Using Change of Variables and Quotient Rule
Replies
5
Views
1K
Curve for a line integral - direction confusion
Replies
9
Views
581
Deduce orthogonality relations for sine and cosine w/ Euler's Formula
Replies
1
Views
572
Line Integral of circle in counterclockwise direction
Replies
5
Views
1K
Find the area delimited by two polar curves
Replies
2
Views
1K
Integral Question: Areas between curves?
Replies
1
Views
1K
How should I find the equilibrium points and the general equation?
Replies
3
Views
313
Outward flux of a vector field
Replies
6
Views
1K
Reparameterizing a curve in path length parameter
Replies
1
Views
1K
Computing line integral using Stokes' theorem
Replies
8
Views
1K

Hot Threads

Recent Insights

Back
Top