Find the area:
f(x)=sin6x and g(x) = cos 12x, -pi/12 less than or euqal to x is less than or equal to pi/36
To find the area between two curves, we need to determine the points of intersection first.
Let's set the two functions f(x) and g(x) equal to each other:
sin(6x) = cos(12x)
To solve this equation, we can use a trigonometric identity: sin(π) = cos(π/2 - π).
Therefore, we have:
sin(6x) = cos(π/2 - 12x)
Comparing the angles, we get:
6x = π/2 - 12x
Let's solve for x:
18x = π/2
x = π/36
The points of intersection are x = π/36.
To find the area between the curves, we integrate the difference of the two functions over the given interval.
β«[a,b] [f(x) - g(x)] dx
Using the formula for the area between two curves, the integral becomes:
Area = β«[-π/12, π/36] [sin(6x) - cos(12x)] dx
To find this integral, we integrate each function separately:
β«[-π/12, π/36] sin(6x) dx = [-1/6cos(6x)] [-π/12, π/36]
= [-1/6cos(6(π/36))] - [-1/6cos(6(-π/12))]
= [-1/6cos(π/6)] - [-1/6cos(-π/2)]
= [-1/6(β3/2)] - [-1/6(0)]
= -β3/12
β«[-π/12, π/36] cos(12x) dx = [1/12sin(12x)] [-π/12, π/36]
= [1/12sin(12(π/36))] - [1/12sin(12(-π/12))]
= [1/12sin(π)] - [1/12sin(-π)]
= [1/12(0)] - [1/12(0)]
= 0
Therefore, the area between the curves f(x) = sin(6x) and g(x) = cos(12x), over the interval -π/12 β€ x β€ π/36, is -β3/12.