Areas and Antiderivatives

Question

First Central City, residential power consumption has shown certain trends over recent years. Based on data reflecting the average usage, scientists at S.T.A.R Labs have modeled the city's rate of energy consumption by the function

r(t) = 4 + sin(0.263t  + 4.7) + cos(0.526t + 9.4)

Here, t measures time in hours after midnight on a typical weekday, and r is the rate of consumption and megawatts at time t. (Note: A megawatt is actually a rate; It measures energy consumption per unit time. A megawatt-hour is the total amount of energy that is equivalent to a constant one megawatt of power over one hour.)

(a)   Without calculating its value, explain the meaning of ∫₀²⁴ _{r(t) dt}

(b)   Determine the amount of power Central City consumes in a day

(c)    What is Central City’s average rate of energy consumption in a given 24-hour period? Indicate unit of measure.

Answer 1

a) This integral represents the total amount of power (in megawatt-hours) used by the city on a typical weekday.

b) Use a calculator or other resource to calculate the integral in a).

c) Divide your answer from b) by 24 hours to get the average rate (in megawatts).