91Ó°ÊÓ

Question: Using the definite integral, find the net change in the quantity Q(t) given its rate of change function R(t) = 3t^2 - 2t from time t = 1 second to t = 4 seconds. Answer: To find the net change in the quantity Q(t) over the interval [1, 4], we need to calculate the definite integral of R(t) = 3t^2 - 2t over the given interval. The definite integral is written as: \[∫_{1}^{4} (3t^2 - 2t) dt\] In this case, we can use the fundamental theorem of calculus to solve the integral. The antiderivative of R(t) = 3t^2 - 2t is: Q'(t) = (3/3)t^3 - (2/2)t^2 + C = t^3 - t^2 + C Now evaluate the definite integral: \[Q(4) - Q(1) = (4^3 - 4^2) - (1^3 - 1^2) = (64 - 16) - (1 - 1) = 48\] The result is positive, indicating an overall increase in the quantity. The net change in the quantity Q(t) over the given interval [1, 4] is 48 units.