I have three functions: $$y=5+\sin x\\ y=-x+0.5\\ y=2x+1$$I have to calculate area between them. I made: $-x+0.5=2x+1$ and I got $x=-\frac16$. What shall I do next to get the area?
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Yes, the two lines intersect at $x= -\frac{1}{6}$. For x< -1/6 you want to integrate $5+ sin(x)- (0.5- x)$. For x> -1/6 you want to integrate $5+ sin(x)- (2x+ 1)$. The hard part is finding the lower and upper bounds on the integral. The lower bound is x satisfying $5+ sin(x)= 0.5- x$ and the upper bound is x satisfying $5+ sin(x)= 2x+ 1$. You will need to use some numerical method to determine those values. – user247327 Nov 21 '17 at 13:27
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You won't find nice solutions to $2x+1 = 5 \sin x$ or $-x+0.5 = 5+\sin x.$ You can get numerical answers, however. The first has solution $x = 2.35.$ The second has solution $x=-5.32.$ You'll have to break the area into two integrals:
$$\int_{-5.32}^{-1/6} (5+\sin x) -(-x+0.5) \; dx + \int_{-1/6}^{2.35} (5+\sin x) -(2x+1) \; dx.$$
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In calculus we always use radians. Otherwise the derivative of $\sin x$ is not $\cos x.$ – B. Goddard Nov 21 '17 at 13:51