Problem 115 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{-1}^{0} e^{3 x} d x $$

Step-by-Step Solution

Answer

\( \frac{e^3 - 1}{3e^3} \)

Step 1: Understand the Problem

We are asked to evaluate the definite integral \( \int_{-1}^{0} e^{3x} \, dx \). This involves finding the area under the curve of the function \( e^{3x} \) from \( x = -1 \) to \( x = 0 \).

Step 2: Find the Antiderivative

The integral of \( e^{3x} \) can be found using the formula \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \). Here, \( k = 3 \), so the antiderivative is \( \frac{1}{3} e^{3x} \).

Step 3: Apply the Limits of Integration

Evaluate the antiderivative at the upper limit (0) and the lower limit (-1) and subtract: \[ \left[ \frac{1}{3} e^{3x} \right]_{-1}^{0} = \frac{1}{3} e^{0} - \frac{1}{3} e^{-3} \].

Step 4: Calculate and Simplify

Compute the values: \[ e^{0} = 1 \, \text{and} \, e^{-3} = \frac{1}{e^3} \]. Thus, the result is:\[ \frac{1}{3} \times 1 - \frac{1}{3} \times \frac{1}{e^3} = \frac{1}{3} - \frac{1}{3e^3} \]. Simplifying the expression gives:\[ \frac{e^3 - 1}{3e^3} \].

Key Concepts

Antiderivative

The antiderivative is a fundamental concept in calculus. It refers to the reverse process of differentiation. Finding an antiderivative means identifying a function whose derivative yields the original function.
In the context of definite integrals, knowing the antiderivative is essential because it helps calculate areas under curves. In this exercise, we seek the antiderivative of the function \( e^{3x} \).
### Formula for Exponential FunctionsFor exponential functions of the form \( e^{kx} \), where \( k \) is a constant, the antiderivative can be found using a simple formula:

• \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \)

Here, "\( C \)" denotes a constant of integration which is usually omitted in definite integrals.
This is because definite integrals are about finding a net area, and the constant cancels out when applying limits. In our case, with \( k = 3 \), the antiderivative becomes \( \frac{1}{3} e^{3x} \).
Thus, understanding and finding antiderivatives marks a vital step in solving calculus problems.

Limits of Integration

The limits of integration in a definite integral specify the interval over which the function is integrated. These limits define the range for which we calculate the area under the graph of the function.
In this exercise, our limits of integration are from \( -1 \) to \( 0 \). This tells us we are interested in the area from \( x = -1 \) to \( x = 0 \) on the curve of \( e^{3x} \).
### Evaluating with LimitsOnce we find the antiderivative, we apply these limits using the Fundamental Theorem of Calculus:

• The antiderivative is evaluated at the upper limit (0).
• Then, it’s evaluated at the lower limit (-1).
• Subtract the lower limit value from the upper limit value.

In mathematical terms, this is represented as:\[\left[ \frac{1}{3} e^{3x} \right]_{-1}^{0} = \frac{1}{3} e^{0} - \frac{1}{3} e^{-3}\]
Thus, limits of integration help us identify precisely where to calculate the net area.

Exponential Function

Exponential functions are a class of functions with the form \( f(x) = a \cdot e^{kx} \), where "\( e \)" is a mathematical constant approximately equal to 2.71828. It's known as Euler's number. These functions model growth or decay processes.
In this exercise, the function \( e^{3x} \) is the integrand. Its exponential nature means rates of change increase rapidly as \( x \) increases.
### Properties of Exponential FunctionsExponential functions have unique qualities:

• The base (\( e \)) is constant.
• The curve never touches the x-axis, reflecting continuous growth or decay.
• The derivative of \( e^{kx} \) keeps an exponential nature, making it simpler to integrate.

Examining these characteristics helps in identifying their antiderivatives. For \( e^{3x} \), integration results in a straightforward antiderivative: \( \frac{1}{3} e^{3x} \).
Process-wise, this characteristic is vital as exponential functions appear frequently in applications such as population growth, interest calculations, and natural processes modeling. Understanding them is crucial for calculus proficiency.