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Find the area under one arch of the curve y=cos x/4

Sagot :

Sakurablossom2idn

Answer:

8

Step-by-step explanation:

First, we get the boundaries of the arch. The boundaries of the arch has y-coordinate of zero (when it crosses the x-axis), thus,

0 = cos(x/4)

cos^-1 (0) = x/4

π/2 = x/4

x = 2π

-π/2 = x/4

x = -2π

Thus, boundary is from -2π to 2π. Actually, there are several boundaries (like from 6π to 10π) if you plot y=cos(x/4), but here, lets just use -2π to 2π.

Then set-up the integral. Imagine vertical strips that fill the region inside the arch, that goes from -2π to 2π. Thus,

∫ y dx , from -2π to 2π

∫ cos(x/4) , from -2π to 2π

= 4 sin(x/4) , from -2π to 2π

= 4 sin(2π/4) - 4 sin(-2π/4)

= 4(1) - 4(-1)

= 4 + 4

= 8

You can also do the same for boundary from 6π to 10π, you should get the same answer.

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