"I would like to integrate the following equation. It is a chemical
rate equation (Arrhenius) where I have temperature & relative humidity
as functions of time. Both are sinusoidal, reflecting a typical yearly
climate":
e^-(Integral(k*RH(t))dt), where k=Ae^-(E/R*T(t)) and
RH(t) = 0.7+0.1*Sin(0.01t), T(t) = 300+4.5*Sin(0.01t).
To integrate the given equation e^-(Integral(k*RH(t))dt), we need to first evaluate the integral of k*RH(t) with respect to t. However, the integrand (k*RH(t)) is dependent on two variables, RH(t) and T(t), which are both functions of time.
In this case, RH(t) and T(t) are sinusoidal functions, where RH(t) = 0.7 + 0.1*sin(0.01t) and T(t) = 300 + 4.5*sin(0.01t).
To integrate this equation, we need to follow these steps:
Step 1: Substitute the given expressions for RH(t) and T(t) into the expression for k.
k = A*e^(-(E/R)*T(t))
= A*e^(-(E/R)*(300 + 4.5*sin(0.01t)))
Step 2: Integrate k*RH(t) with respect to t.
∫(k*RH(t))dt = ∫(A*e^(-(E/R)*(300 + 4.5*sin(0.01t)))*(0.7 + 0.1*sin(0.01t)))dt
This integral is non-trivial and cannot be solved analytically in general. However, you can use numerical methods or software tools like MATLAB, Python (using libraries like scipy), or other mathematical software to evaluate the integral numerically.
The numerical integration methods, such as Simpson's rule or the trapezoidal rule, can be used to approximate the integral. These methods divide the integration range into small intervals, evaluate the function at multiple points within each interval, and sum up the contributions from each interval to approximate the integral.
By using numerical methods, you should be able to obtain an approximate value for the integral of e^-(Integral(k*RH(t))dt) given the specific values for A, E, and the time range over which you wish to evaluate the integral.