Help! Integration Question (1 Viewer)

[tonyharrison]

From the 2007 HSC exam: Question 7, (b) (ii)
http://www.boardofstudies.nsw.edu.au/hsc_exams/exam-papers-2007/pdf_doc/maths-07.pdf


I'm having trouble finding the shaded area between the sin and cos graph because part of the area is above the x-axis, and some is below. As well as this, the two graphs don't cross the x-axis at the same point, so you can't split the integration process into two.


If someone could help it would be much appreciated. THankyou!!

 

[Yaziyo]

You can just do it normally and it will work. Try it as just the integral of sinx - the intragral of cosx between their points of intersection and you'll get the result.

If you want to be safe, you could add a constant to both of them so you're working with sinx + 10 and cosx + 10 or something, which would be the same area just shifted upwards.

 

[tonyharrison]

Sorry i didn't make it very clear, the graphs are y=sinx and y=(root 3)cosxI tried doing it normally, but it just ended up cancelling each other out. THe points of intersection are at (pi/3) and (4pi/3)


Do i need to change the cos or sin to minus because it's in the wrong quadrant or something?

 

[Yaziyo]

You don't need to change anything with it. You might be cancelling out because of the signs.

integrate (sinx - rt3 cosx) between 4pi/3 and pi/3
= [-cosx - rtsinx]
= -[cosx + rtsinx]
= -(cos(4p/3) + rt3 sin (4p/3) - cos(p/3) - rt3 sin(p/3)
= -(-0.5 -1.5 - 0.5 - 1.5)
= 4units^2

 

[tonyharrison]

Oh i see what i did wrong!
Thanks so much.
Haha you're smarter than my maths teacher =D

 

[scumm]

omg, i would've had to butcher the graph in order to find the area. What type of graphs would i need to split into different sections to find the area then? Or can we really just do integral(top curve - bottom curve) for every one of these questions??

 

[bmn]

omg, i would've had to butcher the graph in order to find the area. What type of graphs would i need to split into different sections to find the area then? Or can we really just do integral(top curve - bottom curve) for every one of these questions??

Top-minus should work for all where there are 2 curves. You might need to split it when its 1 line/curve.

 

[Yaziyo]

@tonyharrison - no worries

@scumm - You should be able to do it for any area between 2 curves, because the bits that would cancel when using one curve, uncancel again, if that makes any sense. Think of it this way, if it was the area between (sinx + 10) and (rt3 cosx + 10), it's obviously not going to the cut the x axis, but when you find the area between them, the 10s will cancel out, giving you the same number. But obviously if it was just the area under the curve (e.g just finding the area under sinx) you can't just randomly add numbers