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Let F(a) be the area between the x-axis and the graph of y=x^2cos(x/4) between x=0 and x=a, for a>0 (consider the area to be negative if the graph lies below the axis).
Find the formula for F(a).
what was the answer
To find the formula for F(a), we need to integrate the given function y = x^2cos(x/4) with respect to x over the interval [0, a].
First, let's write down the integral:
F(a) = ∫[0, a] (x^2cos(x/4)) dx
To solve this integral, we can use integration by parts. The general formula for integration by parts is:
∫u v dx = uv - ∫v du
In this case, we can choose u = x^2 and dv = cos(x/4) dx.
To find du, we differentiate u with respect to x:
du = d/dx (x^2) dx = 2x dx
To find v, we integrate dv with respect to x:
v = ∫ cos(x/4) dx
We can evaluate this integral by applying a substitution. Let u = x/4, then du = 1/4 dx. Substituting back, we have:
v = ∫ cos(u) (4 du) = 4 ∫ cos(u) du
Integrating cos(u) gives us sin(u):
v = 4 sin(u)
Substituting x/4 back in for u, we have:
v = 4 sin(x/4)
Now we can use the integration by parts formula to solve the integral:
F(a) = ∫[0, a] (x^2cos(x/4)) dx
= [u v] - ∫[0, a] v du
= [x^2 * 4 sin(x/4)] - ∫[0, a] (4sin(x/4) * 2x) dx
= 4x^2 sin(x/4) - 8 ∫[0, a] x sin(x/4) dx
The second integral on the right-hand side represents a new integral that needs to be solved. To evaluate this integral, we can use integration by parts again.
Let u = x, and dv = sin(x/4) dx.
Then, du = dx, and v = -4 cos(x/4).
Using the integration by parts formula again, we have:
∫[0, a] x sin(x/4) dx
= [uv] - ∫[0, a] v du
= [-4x cos(x/4)] - ∫[0, a] (-4 cos(x/4)) dx
= -4x cos(x/4) + 4 ∫[0, a] cos(x/4) dx
= -4x cos(x/4) + 16 sin(x/4)
Now we can substitute this back into the expression for F(a):
F(a) = 4a^2 sin(a/4) - 8 (-4a cos(a/4) + 16 sin(a/4))
= 4a^2 sin(a/4) + 32a cos(a/4) - 128 sin(a/4)
Therefore, the formula for F(a) is:
F(a) = 4a^2 sin(a/4) + 32a cos(a/4) - 128 sin(a/4)
We integrate by parts twice:
1)u=x^2, du=2xdx, dv=cos(x/4)dx, v=4sin(x/4)
2)u=8x, du=8dx, dv=sin(x/4)dx, v=-4cos(x/4)
The answer I have already sent.