Problem 11 - Applied Calculus

Question

Solar photovoltaic (PV) cells are the world's fastestgrowing energy source. \({ }^{7}\) Annual solar \(\mathrm{PV}\) production, \(S\), in megawatts, is approximated by \(S=277 e^{0.368 t}\), where \(t\) is in years since 2000 . Estimate the total solar PV production between 2000 and 2010 .

Step-by-Step Solution

Answer

The total solar PV production between 2000 and 2010 is approximately 28,938 megawatts.

Step 1: Define the Problem

We need to find the total solar PV production from the year 2000 to 2010. This equates to calculating the integral of the function \( S = 277 e^{0.368 t} \) from \( t = 0 \) to \( t = 10 \).

Step 2: Set Up the Integral

To find the total production, set up the integral of the equation from \( t = 0 \) to \( t = 10 \): \[ \int_{0}^{10} 277 e^{0.368 t} \, dt \].

Step 3: Calculate the Integral

We will solve the integral using the formula \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( a = 0.368 \).Thus, the integral becomes:\[ \int 277 e^{0.368 t} \, dt = 277 \times \frac{1}{0.368} e^{0.368 t} \].

Step 4: Evaluate the Definite Integral

The solution to the previous step gives us \[ \frac{277}{0.368} e^{0.368 t} \].Now, evaluate this from \( t = 0 \) to \( t = 10 \):\[ \frac{277}{0.368} (e^{0.368 \times 10} - e^{0}) \].

Step 5: Compute the Values

Calculate the values:\( e^{0.368 \times 10} = e^{3.68} \approx 39.47 \)So,\[ \frac{277}{0.368} \times (39.47 - 1) \approx \frac{277}{0.368} \times 38.47 \].

Step 6: Final Calculation

Compute the final result:\[ \frac{277}{0.368} \approx 752.72 \]\[ 752.72 \times 38.47 \approx 28937.88 \].Thus, the estimated total production is approximately 28,938 megawatts.

Key Concepts

Exponential Growth

Exponential growth describes a process where a quantity increases over time. This can happen when the growth rate is proportional to the current value, which means the bigger the value gets, the faster it grows. In mathematical terms, its formula is generally expressed as \( y = a e^{bx} \), where \( a \) is the initial value, \( e \) is the base of the natural logarithm, \( b \) is the growth rate, and \( x \) is time.
For solar PV production, it is modeled by \( S = 277 e^{0.368t} \).

Key Characteristics of Exponential Growth:

• Rapid Increase: Over time, the value doesn't just grow steadily, it can start increasing very rapidly.
• Constant Percentage Rate: The growth is at a constant percentage rate rather than a fixed amount.
• Common in Nature: Many natural processes like population growth or radioactive decay follow an exponential pattern.

Understanding this pattern in solar PV energy shows us how technologies can quickly become dominant and widespread once growth kicks in.

Solar Photovoltaic Production

Solar photovoltaic (PV) production harnesses sunlight to generate electricity. It is seen as a sustainable and renewable energy source with minimal negative environmental impact. The process involves converting sunlight into electrical energy using photovoltaic cells.

Benefits of Solar PV Production:

• Clean Energy: Produces electricity without greenhouse gases.
• Renewable Source: Uses an inexhaustible power source – the sun.
• Cost-Effective: Long-term, solar PV systems have low operational costs after initial installation.
• Energy Independence: Helps reduce reliance on imported fuels.

Understanding the significance of solar PV production's rapid expansion helps us appreciate shifts in global energy policies towards cleaner options. Massive exponential growth seen in its production underscores solar energy potential as a major future energy source.

Definite Integral Calculation

In calculus, a definite integral is a way to calculate the area under a curve over a specific interval, effectively summing up small quantities to find the total quantity. For a function \( f(x) \), the definite integral from \( a \) to \( b \) is represented as \( \int_a^b f(x) \,dx \).
This is a crucial tool in determining total quantities like solar PV production over time.

Steps to Calculate a Definite Integral:

• Identify Function: Begin by determining the function you need to integrate, e.g., \( S = 277 e^{0.368t} \).
• Set Limits: Choose the integration limits, \( a = 0 \) and \( b = 10 \) for 2000 to 2010.
• Integrate: Use integration rules to find the antiderivative of the function.
• Evaluate: Substitute the upper and lower limits into the antiderivative and calculate the difference.

This process allows us to determine the total solar PV production from 2000 to 2010, revealing how calculus tools like integration help quantify real-world phenomena.