Textbook Question
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ 3^{-2x} dx
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11. Integrals of Inverse, Exponential, & Logarithmic Functions
Integrals of Exponential Functions
8:05 minutesProblem 7.2.24b
Textbook Question
Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.
b. Find the total energy (in MW-yr) used by the city over four full years beginning at t=0.
Verified step by step guidance1
Identify the given information: the initial rate of electricity consumption at time \(t=0\) is \(R_0 = 2000\) MW, and the rate increases at a rate of 1.3% per year. This means the rate of consumption grows exponentially.
Express the rate of electricity consumption as a function of time \(t\) (in years). Since the rate increases by 1.3% per year, the rate function is \(R(t) = 2000 \times (1.013)^t\) MW.
To find the total energy used over four years, we need to integrate the rate function over the interval from \(t=0\) to \(t=4\). The total energy \(E\) is given by the integral \(E = \int_0^4 R(t) \, dt = \int_0^4 2000 \times (1.013)^t \, dt\).
Set up the integral for the exponential function. Recall that the integral of \(a^t\) with respect to \(t\) is \(\frac{a^t}{\ln(a)}\), where \(a > 0\) and \(a \neq 1\). So, the integral becomes \(2000 \times \int_0^4 (1.013)^t \, dt = 2000 \times \left[ \frac{(1.013)^t}{\ln(1.013)} \right]_0^4\).
Evaluate the definite integral by substituting the limits \(t=4\) and \(t=0\) into the antiderivative expression, then subtract the lower limit value from the upper limit value to find the total energy consumed over the four years.
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Video duration:
8mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Growth
Exponential growth describes a quantity that increases at a rate proportional to its current value. In this problem, the electricity usage rate grows by 1.3% per year, meaning the rate at time t can be modeled as an exponential function, R(t) = R_0 * e^(kt), where k is the growth rate.
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Integration to Find Total Accumulated Quantity
To find the total energy used over a time interval, we integrate the rate function over that period. Integration sums the instantaneous rates over time, giving the total accumulated energy consumption in MW-years.
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Converting Percentage Growth to Continuous Growth Rate
A percentage growth rate given per year (1.3%) can be converted to a continuous growth rate k by using k = ln(1 + r), where r is the decimal form of the percentage. This allows modeling the growth with the exponential function e^(kt) for continuous compounding.
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