Final answer:
To evaluate the integral ∫10r(t) dt where r(t) has two different definitions, you need to perform the integral in two parts, one for each definition of r(t), and then sum the results to get the final answer.
Step-by-step explanation:
To evaluate the integral of ∫10r(t) dt, where r(t) is a function of t, you need to first identify the function r(t) correctly. However, the question as it stands seems to contain a typographical error in specifying the function r(t). Assuming r(t) = 3t when t ≤ 1 and r(t) = e^(-t) otherwise, the integral would be broken into two parts.
For t ≤ 1:
∫^{1}_{0} 3t dt = [1.5t^2] evaluated from 0 to 1 = 1.5.
For t > 1:
∫^{10}_{1} e^(-t) dt = [-e^(-t)] evaluated from 1 to 10 ≈ -0.000045 + e^(-1).
Adding these results gives the final integral value, which would include the sum of 1.5 and the value obtained from the exponential portion of the integral.