Trig curves and integration. (1 Viewer)

[girlworld_club]

I keep stumbling on trig curves where i am asked to find the area between 2 curves e.g sinx and -cosx between 2 certain points. In the solutions they always subtract one curve from the other. How do you know which one to subtract from the other to find the area shaded, there is nothing in my textbook and i don't know what to consolidate. If you can give me examples with your explanation that would be heaps heaps appreciated.

 

[bedpotato]

Top curve - bottom curve.



For this example, if you were asked to find the area between y = sinx and y = cos x, between x = 0 and the the first point of intersection, it would be cosx - sinx.
If it was between the first time they intersect and the second time they intersect, sinx is now on top, so sinx - cosx.

The limits are the x values.

 

[girlworld_club]

i am still a bit confused about the top curve-lower curve theory. How do you know which one is "on top".

 

[HeroicPandas]

i am still a bit confused about the top curve-lower curve theory. How do you know which one is "on top".

Common sense

y = x^2 + 2
y = x^2

Which one is the top curve? i.e. which graph within the domain given/found will appear on top (or having more larger y-values for each corresponding x-value)

 

[bedpotato]

i am still a bit confused about the top curve-lower curve theory. How do you know which one is "on top".

It might be easier to just draw a sketch of the two curves if you can (for the question you asked with sinx and -cosx, it's not hard to draw), and visually see which curve is on the top/bottom. Or do what HeroicPandas suggested if the curve is too hard to draw/takes too long to draw.

 

[girlworld_club]

when i draw the graph, as you stated (first post) it will be cosx-sinx

However when i try to think of it logically, ie. draw the cos x graph sepeartly and then minus it from the sinx between the 0 and the (pie)/2 the areas seem to cancel out each other because they are exactly the same.

 

[bedpotato]

You don't minus the cosx graph from the sinx graph.

Let's take your example:
"find the area between 2 curves e.g sinx and -cosx between 2 certain points"
So first draw the sinx graph as you normall would, and then draw the -cosx graph on the same plane as you normally would draw a -cosx graph. You don't minus anything from anything else. Draw two graphs (sinx and -cosx, or whatever they may be) on the same number plane.

Like the sinx and cosx graphs that are attahced in my first post.

 

[girlworld_club]

and how would you find the area, sorry for being a bug but it's hard to understand.

 

[HeroicPandas]

It might be easier to just draw a sketch of the two curves if you can (for the question you asked with sinx and -cosx, it's not hard to draw), and visually see which curve is on the top/bottom. Or do what HeroicPandas suggested if the curve is too hard to draw/takes too long to draw.

nah lol, it was my way of explaining to her which would be the top curve if u had the graph (or drew it)

 

[bedpotato]

Lol, sorry if my explanations aren't that great.


If the question asked to find the shaded area (i.e. between x = 0 and x = A [let A = pi/3]). Which graph is on top between x = 0 and x = pi/3? cosx is, and the sinx is at the bottom. So to find the area:



And then just integrate normally and so on.

 

[bedpotato]

nah lol, it was my way of explaining to her which would be the top curve if u had the graph (or drew it)

Lol oh. For some reason I thought you meant that you should sub x values into both curves and whichever one has a higher y-value is the one on top.

 

[HeroicPandas]

and how would you find the area, sorry for being a bug but it's hard to understand.

before anything happens, forget about that weird "top curve-lower curve theory"
For ,

graph y = sinx and y = -cosx

If u wanna find the area between these curves, get a blue and red pen. Colour the area under the curve y= sinx blue, and the area under the curve y = -cosx red


Now apply integration:





Now A(red) is where people get tricked, since the area is under the x-axis, u must put absolute values around the whole integration or else ur answer will be negative








This is the same as going top curve minus bottom curve

 

[HeroicPandas]

Lol oh. For some reason I thought you meant that you should sub x values into both curves and whichever one has a higher y-value is the one on top.

yeh it does haha, sorry, my lack of clearness caused ambiguity

 

[Symphonicity]

Sketch the two curves and look at the graph between the two points you are trying to find the area between.. Whichever one is higher up on the y axis is the one you want to write first, then subtract the other curve which will be lower on the y axis.