Textbook Question
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀ˡⁿ ² (e^{3x} − e^{−3x}) / (e^{3x} + e^{−3x}) dx
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11. Integrals of Inverse, Exponential, & Logarithmic Functions
Integrals of Exponential Functions
7:26 minutesProblem 7.2.24c
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.
c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.
Verified step by step guidance1
Identify the given information: the initial rate of electricity consumption at time \(t=0\) is 2000 MW, and the rate increases by 1.3% per year. This suggests the rate function grows exponentially.
Express the rate of electricity consumption as a function of time \(t\). Since the rate increases by 1.3% per year, the rate function can be written as \(R(t) = 2000 \times (1.013)^t\), where \(t\) is in years.
Understand that the total energy used between \(t=0\) and any future time \(t\) is the integral of the rate function over that interval. So, the total energy function \(E(t)\) is given by the integral \(E(t) = \int_0^t R(x) \, dx\).
Set up the integral explicitly: \(E(t) = \int_0^t 2000 \times (1.013)^x \, dx\). This integral will give the total energy consumed in megawatt-years (MW-yr) from time 0 to time \(t\).
To solve the integral, recall that the integral of an exponential function \(a^x\) with respect to \(x\) is \(\frac{a^x}{\ln(a)}\). Use this formula to express \(E(t)\) in terms of \(t\) without calculating the final numeric value.
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Video duration:
7mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Growth
Exponential growth describes a process where a quantity 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 function can be modeled as an exponential function of time.
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Integration of a Rate Function
To find the total quantity accumulated over time from a rate function, we integrate the rate with respect to time. Here, integrating the power consumption rate over the interval [0, t] gives the total energy used in megawatt-years.
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Units and Interpretation of the Integral
Understanding units is crucial: the rate is in megawatts (MW), and integrating over time (years) yields energy in megawatt-years (MW-yr). This helps interpret the integral as total energy consumed, not just instantaneous power.
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