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 a formula for F(a).
To find a formula for F(a), we need to integrate the function y=x^2cos(x/4) with respect to x over the interval [0, a].
Let's start by finding the indefinite integral of the function, which will give us an antiderivative. Notice that the integrand involves both x^2 and cos(x/4), so we can use the technique of integration by parts.
Integration by parts states that for two functions, u(x) and v(x), the integral of their product can be evaluated as the product of one function evaluated at the limits of integration and the integral of the other function, minus the integral of the derivative of the one function multiplied by the integral of the other function.
In this case, we can choose u(x) = x^2 and dv(x) = cos(x/4) dx. Then, we can find the derivative of u(x), which is du(x) = 2x dx, and the integral of dv(x), which is v(x) = 4sin(x/4).
Using the formula for integration by parts, we have:
∫ x^2 cos(x/4) dx = x^2 * 4sin(x/4) - ∫ (2x * 4sin(x/4)) dx
Simplifying the above equation, we have:
∫ x^2 cos(x/4) dx = 4x^2 sin(x/4) - 8∫ x sin(x/4) dx
Now, we have a new integral to evaluate: ∫ x sin(x/4) dx. We can use integration by parts again to solve this integral.
Let's choose u(x) = x and dv(x) = sin(x/4) dx. Then, we have du(x) = dx and v(x) = -16cos(x/4).
Applying the formula for integration by parts again, we get:
∫ x sin(x/4) dx = -16x cos(x/4) - ∫ (-16 cos(x/4)) dx
Simplifying further, we have:
∫ x sin(x/4) dx = -16x cos(x/4) + 16∫ cos(x/4) dx
The integral of cos(x/4) is straightforward: ∫ cos(x/4) dx = 4sin(x/4).
Now, we can substitute the results back into the previous equation:
∫ x sin(x/4) dx = -16x cos(x/4) + 16(4sin(x/4))
Simplifying the equation:
∫ x sin(x/4) dx = -16x cos(x/4) + 64sin(x/4)
Now, we can substitute this result back into the previous equation for the integration of x^2 cos(x/4):
∫ x^2 cos(x/4) dx = 4x^2 sin(x/4) - 8(-16x cos(x/4) + 64sin(x/4))
Simplifying this equation, we get:
∫ x^2 cos(x/4) dx = 4x^2sin(x/4) + 128x cos(x/4) - 512sin(x/4)
Finally, to find the formula for F(a), we evaluate this antiderivative at the upper limit a, and subtract the value at the lower limit 0:
F(a) = 4a^2 sin(a/4) + 128a cos(a/4) - 512sin(a/4) - (0)
Simplifying the formula for F(a):
F(a) = 4a^2 sin(a/4) + 128a cos(a/4) - 512sin(a/4)
Therefore, the formula for F(a) is 4a^2 sin(a/4) + 128a cos(a/4) - 512sin(a/4).