Question

The diagram shows two curves with equations y=sin x and y=sin2x for x values between 0 and pi. the curves meet at the origin and at the points P and Q. a) find P and Q? b)find the areas of the shaded regions A1 and A2?

Answer 1

We need to set the two curves equal.

`sinx = sin(2x)`

`0 = sin(2x) - sinx`

`0 = 2sinxcosx- sinx`

`0 = sinx(2cosx- 1)`

`sinx= 0 or cosx = 1/2`

`x = 0, pi, pi/3`

Now we set up our integral expressions for area.

`A_1 = int_0^(pi/3) sin(2x) - sinx dx`

`A_1 = [-1/2cos(2x) + cosx]_0^(pi/3)`

`A_1 = -1/2cos((2pi)/3) + cos(pi/3) + 1/2cos(0) - cos(0)`

`A_1 = 1/4 + 1/2 + 1/2 - 1`

`A_1 = 1/4` square units.

Now onto `A_2`.

`A_2 = int_(pi/3)^pi sinx - sin(2x)dx`

`A_2 = [ -cosx + 1/2cos(2x)]_(pi/3)^pi`

`A_2 = -cos(pi) + 1/2cos(2pi) - (-cos(pi/3) + 1/2cos(2(pi/3))`

`A_2 = 1 + 1/2 + 1/2 + 1/4`

`A_2 = 9/4`

The total area is `5/2` square units.

Hopefully this helps!