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

The evaluated integral is \(\frac{1}{3}(1 - e^{-3})\).

Step 1: Set Up the Problem

To evaluate the definite integral \( \int_{-1}^{0} e^{3x} \, dx \), identify the limits of integration \(-1\) and \(0\), and recognize that you are integrating the function \(e^{3x}\).

Step 2: Find the Antiderivative

The function \(e^{3x}\) has an antiderivative of \(\frac{1}{3}e^{3x}\), because the derivative of \(\frac{1}{3}e^{3x}\) is \(e^{3x}\). This follows from the chain rule, where differentiating \(e^{3x}\) gives \(3e^{3x}\), and multiplying by \(\frac{1}{3}\) cancels the 3.

Step 3: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the definite integral from \(a\) to \(b\) of a function is the difference between its antiderivative evaluated at \(b\) and \(a\). For this integral, evaluate \(\frac{1}{3}e^{3x}\) at \(x=0\) and \(x=-1\).

Step 4: Evaluate at the Upper Limit

Calculate the antiderivative at the upper limit: \(\frac{1}{3}e^{3 \times 0} = \frac{1}{3}e^{0} = \frac{1}{3}\), since \(e^{0} = 1\).

Step 5: Evaluate at the Lower Limit

Calculate the antiderivative at the lower limit: \(\frac{1}{3}e^{3 \times (-1)} = \frac{1}{3}e^{-3}\).

Step 6: Subtract to Find the Result

Subtract the value at the lower limit from the value at the upper limit: \(\frac{1}{3} - \frac{1}{3}e^{-3}\).

Step 7: Simplify the Expression

The expression \(\frac{1}{3} - \frac{1}{3}e^{-3}\) can be factored as \(\frac{1}{3}(1 - e^{-3})\). This is the final evaluated integral.

Key Concepts

Antiderivative

An antiderivative is essentially the reverse of taking a derivative. It involves finding a function whose derivative is the function given. For the exponential function \(e^{3x}\), we find its antiderivative by considering what function would, when differentiated, give us \(e^{3x}\). In this case, the answer is \(\frac{1}{3}e^{3x}\).
When you differentiate \(\frac{1}{3}e^{3x}\) with respect to \(x\), you multiply by the inner derivative of \(3x\), which introduces a 3, resulting in \(e^{3x}\). Thus, the correct antiderivative is \(\frac{1}{3}e^{3x}\). This understanding helps us simplify complex calculus problems by reversing the differentiation process.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of an integral to an antiderivative. There are two parts to this theorem, but the one used here states that if you can find an antiderivative of a function, you can use it to evaluate its definite integral between two points.
For the integral \(\int_{-1}^{0} e^{3x} \, dx\), we find the antiderivative, \(\frac{1}{3}e^{3x}\). According to the theorem, we then evaluate this antiderivative at the upper limit (0) and lower limit (-1), subtracting the latter from the former. This approach simplifies the evaluation of integrals significantly and provides a practical application of the derivative-inverse relationship.

Chain Rule

The chain rule is a vital rule in calculus used when differentiating composite functions. It provides a way to find the derivative of a function composed of another function, such as \(e^{3x}\).
To see the chain rule at work in this problem, consider that if you were asked to differentiate \(\frac{1}{3}e^{3x}\), the chain rule would be applied. Differentiating \(e^{3x}\) directly gives \(3e^{3x}\) (since the derivative of \(3x\) with respect to \(x\) is 3), and then multiplying by \(\frac{1}{3}\) cancels out the 3, resulting in \(e^{3x}\), which confirms that \(\frac{1}{3}e^{3x}\) is the correct antiderivative.

Exponential Functions

Exponential functions, like $e^{3x}$, are a key part of calculus and mathematics as a whole. Their growth is rapid, and they occur frequently in natural phenomena. The function $e^{x}$ is particularly noteworthy because its derivative is itself, making calculations with $e^{x}$ straightforward. For the function $e^{3x}$, the $3x$ in the exponent modifies the rate of growth or decay. The presence of $3x$ means any change in $x$ is amplified by a factor of 3 in terms of growth rate. This characteristic is crucial when considering the scaling effects in various applications, such as compound interest, population models, and radioactive decay.