The curves y=sinx and y=cosx intersects twice on the interval (0,2pi). Find the area of the region bounded by the two curves between the points of intersection.
Answer A=0
To find the area of the region bounded by the curves y = sinx and y = cosx between the points of intersection, we need to first find the x-coordinates of the points of intersection.
To do this, we can set the two equations equal to each other and solve for x:
sinx = cosx
Divide both sides by cosx:
tanx = 1
Now, we need to find the values of x on the interval (0, 2π) that satisfy this equation.
The value of x that solves tanx = 1 in this interval is π/4, which is the first point of intersection.
To find the second point of intersection, we need to find the next value of x where tanx = 1.
Since tangent has a period of π, we can add π to the first point of intersection:
x = π/4 + π = 5π/4
So the second point of intersection is at x = 5π/4.
Now that we have the x-coordinates of the points of intersection, we can find the area of the region bounded by the two curves using the formula for finding the area between two curves:
Area = ∫[a,b] (f(x) - g(x)) dx
where a and b are the x-coordinates of the points of intersection, and f(x) and g(x) are the equations of the curves.
In this case, a = π/4 and b = 5π/4, and the curves are y = sinx and y = cosx.
So the area is:
Area = ∫[π/4, 5π/4] (sinx - cosx) dx
Integrating this expression will give us the area of the region bounded by the two curves between the points of intersection.