Calculus 2, Lec 3A: How to Think About Integrals Conceptually (Flow Rate Example, Linearity)
• Calculus 2 Exam 1 Review Problems and... . Suppose the inflow into a tank is f(t) = 3cos(2t) L/min and the outflow is g(t) = 4sin(3t) L/min. Then the net flow is h(t) = f(t) - g(t) = 3cos(2t) - 4sin(3t). The change in volume ΔV is given by a "conceptual equation" to be the integral of dV = h(t)dt. That is, ΔV = ∫dV = ∫h(t)dt, where dt is an infinitesimal amount of time elapsed. The Fundamental Theorem of Calculus helps us evaluate this integral to find the change in volume. We also illustrate the linearity property of definite integrals in this context and write the general equation.
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