Find area between y=Sin x and y=Cos x

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Homework Help Overview

The discussion revolves around finding the area between the curves defined by the functions y = sin(x) and y = cos(x) over the interval [π/4, 15π/4]. Participants explore the implications of the functions' intersections and the behavior of the area calculation.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of integrals to calculate the area, with one suggesting the need to consider the absolute values to avoid negative areas. There are inquiries about the points of intersection and concerns regarding areas potentially canceling out when one function is above and the other below the x-axis. Some participants mention the periodic nature of the sine and cosine functions and the need to visualize the graphs to avoid double counting areas.

Discussion Status

The discussion is active, with various approaches being explored. Some participants have provided general guidance on using integrals and the importance of identifying points of intersection, while others are questioning the assumptions about area cancellation and the interpretation of the problem's requirements.

Contextual Notes

There is mention of the need to break the area into discrete sections based on the points of intersection, and the discussion reflects uncertainty about how to handle areas that may cancel out. Participants are also considering the implications of the phrasing of the original question regarding whether absolute or differential areas are required.

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Find area between y=Sin x and y=Cos x on intervals [Pi/4, 15Pi/4].
 
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Area can be found by [tex]\int_{x(0)}^{x(1)} (F(x)-G(x))dx[/tex]

Simply let one of the functions be F(x) and the other be G(x). The absolute value of your integral over your limits will be the area (we don't owe the universe area if it comes out negative).

Also, use symetry to your advantage here. from pi/4 to 5pi/4 is one area block. You have 3.75 area blocks thus you can say 3xOne area block plus the area of the last block (13pi\4 to 15pi\4)

good luck.
 
Last edited:
how do you that the areas won't cancel out? If one is above and the other below the x-axis?
 
I can do the intregal part i am just having trouble finding the points of intersection. I know there are 4 different intregals that need to be used.
 
sin and cos functions intersect every (x)pi/4
 
garytse86 said:
how do you that the areas won't cancel out? If one is above and the other below the x-axis?

The answer here depends on how the question is actually phrased. I gave a general response to finding an area using integrals; however, the usage depends on the question. If the absolute area is desired then you must break the function into discrete areas and sum the individual areas. If the question wants the differential area then simple application of the area formula suffices.
 
when u talk about differential area do you mean just normal intergration with limits regardless of "cancelling the area"?
 
Integration is just a convenient way to find the area; however, we cannot always use pure integration in finding out the area. The best way to visualise is to draw the graph. You will see that there will be some common areas between this 2 equation. The trick is that we must avoid adding them twice. By finding out the points of intersection, we can eventually find the area.
 
To find the point of intersection
sin x = cos x
sin x = sqrt(1-(sin x)^2)
then continue the working and you should be able to find all points of intersection.
 

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