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?

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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?

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Answer 1 · 2018-06-11T03:34:22

We need to set the two curves equal.

$sin x = sin (2 x)$ $sinx = sin(2x)$

$0 = sin (2 x) - sin x$ $0 = sin(2x) - sinx$

$0 = 2 sin x cos x - sin x$ $0 = 2sinxcosx- sinx$

$0 = sin x (2 cos x - 1)$ $0 = sinx(2cosx- 1)$

$sin x = 0 or cos x = \frac{1}{2}$ $sinx= 0 or cosx = 1/2$

$x = 0, π, \frac{π}{3}$ $x = 0, pi, pi/3$

Now we set up our integral expressions for area.

$A_{1} = \int_{0}^{\frac{π}{3}} sin (2 x) - sin x d x$ $A_1 = int_0^(pi/3) sin(2x) - sinx dx$

$A_{1} = {[- \frac{1}{2} cos (2 x) + cos x]}_{0}^{\frac{π}{3}}$ $A_1 = [-1/2cos(2x) + cosx]_0^(pi/3)$

$A_{1} = - \frac{1}{2} cos (\frac{2 π}{3}) + cos (\frac{π}{3}) + \frac{1}{2} cos (0) - cos (0)$ $A_1 = -1/2cos((2pi)/3) + cos(pi/3) + 1/2cos(0) - cos(0)$

$A_{1} = \frac{1}{4} + \frac{1}{2} + \frac{1}{2} - 1$ $A_1 = 1/4 + 1/2 + 1/2 - 1$

$A_{1} = \frac{1}{4}$ $A_1 = 1/4$ square units.

Now onto $A_{2}$ $A_2$ .

$A_{2} = \int_{\frac{π}{3}}^{π} sin x - sin (2 x) d x$ $A_2 = int_(pi/3)^pi sinx - sin(2x)dx$

$A_{2} = {[- cos x + \frac{1}{2} cos (2 x)]}_{\frac{π}{3}}^{π}$ $A_2 = [ -cosx + 1/2cos(2x)]_(pi/3)^pi$

$A_{2} = - cos (π) + \frac{1}{2} cos (2 π) - (- cos (\frac{π}{3}) + \frac{1}{2} cos (2 (\frac{π}{3}))$ $A_2 = -cos(pi) + 1/2cos(2pi) - (-cos(pi/3) + 1/2cos(2(pi/3))$

$A_{2} = 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{4}$ $A_2 = 1 + 1/2 + 1/2 + 1/4$

$A_{2} = \frac{9}{4}$ $A_2 = 9/4$

The total area is $\frac{5}{2}$ $5/2$ square units.

Hopefully this helps!

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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?

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