Problem 4 - Calculus Early Transcendentals

Question

Explain how to use definite integrals to find the net change in a quantity, given the rate of change of that quantity.

Step-by-Step Solution

Answer

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.

Step 1: Understand the concept of definite integrals and net change

A definite integral represents the signed area between the curve of a function and the x-axis over a given interval. When it comes to a rate of change function, the definite integral of that function tells us the net change in the original quantity over a specified period. In other words, if we have a function R(t) representing the rate of change of some quantity at time t, then the definite integral of R(t) will tell us the net change in that quantity.

Step 2: Identify the rate of change function and the interval

First, identify the given rate of change function, R(t). The function should be given as either a formula or possibly a table of values for each time t. Next, determine the time interval over which the net change is being calculated, which will be represented as [a, b] where a and b are the starting and ending times respectively.

Step 3: Set up the definite integral

To find the net change in a quantity, given the rate of change of that quantity, write the definite integral of the rate function, R(t), with respect to time(t) over the interval [a, b]. This will appear as follows: \[∫_{a}^{b} R(t) dt\]

Step 4: Calculate the definite integral

To calculate the definite integral, use the fundamental theorem of calculus, numerical integration techniques, or a definite integral table or calculator, depending on the information provided and complexity of R(t). The result will represent the net change in the quantity over the specified interval.

Step 5: Interpret the result

Check the sign of the result that you got and its magnitude. A positive result indicates an overall increase in the quantity, whereas a negative result indicates an overall decrease. The magnitude represents the amount of net change in the quantity over the given interval.