Understand the fundamental theorem of calculus, ∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎), and use it to calculate definite integrals.

Determine \(\int_1^3 (3x^2 + 2x) \, dx\)

A manufacturing plant measures the rate of production of widgets, r(t), in units per hour, where t is the time in hours after the start of a shift. The rate function is modelled by r(t) = 45 + 12t - t² for 0 ≤ t ≤ 8. Determine the total number of widgets produced during the 8-hour shift using the fundamental theorem of calculus.

Consider the function $f(x) = 3x^2 - 4x + 1$. Determine the exact value of $\int_{1}^{4} f(x) \, dx$.

A company models the rate of change of profit, in thousands of dollars per month, using the function \( P'(t) = 3t^2 - 8t + 5 \), where \( t \) is the time in months since the start of the year, \( 0 \leq t \leq 12 \). Calculate the total change in profit between the end of month 2 and the end of month 5.