integration

Question

 

a) Sketch the curve 𝑦=𝑎 𝑐𝑜𝑠2𝑥−𝑎 for 0≤𝑥≤𝜋 showing the intercepts with the 𝑥 and 𝑦 axes and the coordinates of any turning points.

b) Showing all your working carefully find an exact value for the area enclosed between

𝑦=𝑎 𝑐𝑜𝑠2𝑥−𝑎, and the y axis and for 0≤𝑥≤𝜋

(11 marks)

 

Could someone explain to me part b??? I am not sure what you mean by 'the curve enclosed between

𝑦=𝑎 𝑐𝑜𝑠2𝑥−𝑎, and the y axis and for 0≤𝑥≤𝜋'

 

I know there is an area enclosed with the x axis but not sure about the y-axis

 

Could someone explain to me please?

Answer 1

Your best bet is to estimate the area with a right triangle...

 

If you graph it on desmos.com, using a slider for constant A,

you will see the max occurs at (pi/2, -2A)


So the area between the y-axis and the function 
can be estimated as a right triangle.

You can get the area of the triangle with vertices

(0,0) (0,-2a) and (pi/2, -2a)

as

base = pi/2
height = |2a|

So (1/2)(pi/2)|2a| = (1/2) pi * |a|

 

 

--------------------------------------------

 

Your other option is to invert the function and integrate it with respect to y instead of x.

 

You will get a messy inverse trig function containing a square root becuase

 

cos2x = 2cos^2-1 = cos^2 - sin^2 = 1 - 2sin^2

 

No guarantee that can be integrated except numerically....