Problem 33 - Calculus for Biology and Medicine

Question

Determine the constant \(c\) so that $$ \int_{0}^{\infty} c e^{-3 x} d x=1 $$

Step-by-Step Solution

Answer

The constant \(c\) is 3.

Step 1: Understanding the Problem

We need to find the constant \(c\) such that the integral \(\int_{0}^{\infty} c e^{-3x} \, dx = 1\). This means the area under the curve \(c e^{-3x}\) from 0 to infinity should be equal to 1.

Step 2: Setting Up the Integral

The integral is given by \(\int_{0}^{\infty} c e^{-3x} \, dx\). We want to solve this integral and set it equal to 1, in order to find \(c\).

Step 3: Performing the Integration

To solve \(\int_{0}^{\infty} c e^{-3x} \, dx\), first consider the integral \(\int e^{-3x} \, dx\), which evaluates to \(-\frac{1}{3} e^{-3x}\).

Step 4: Applying Limits to the Integral

Now plug in the limits 0 to \(\infty\):\[\int_{0}^{\infty} c e^{-3x} \, dx = c \left[ -\frac{1}{3} e^{-3x} \right]_{0}^{\infty} = c \left( 0 - \left(-\frac{1}{3}e^{0}\right)\right) = c \left(\frac{1}{3}\right)\]

Step 5: Solving for the Constant

Set the evaluated integral equal to 1:\[c \left(\frac{1}{3}\right) = 1\]Solve for \(c\) by multiplying both sides by 3:\[c = 3\]

Step 6: Conclusion

The constant \(c\) is 3. This means when \(c = 3\), the integral \(\int_{0}^{\infty} c e^{-3x} \, dx\) equals 1, satisfying the condition.

Key Concepts

Definite Integrals

Definite integrals are a vital concept in calculus, representing the net area under a curve over a specific interval. Unlike indefinite integrals, which yield a general form of the antiderivative, definite integrals result in a numerical value. This value corresponds to the area between the curve and the x-axis over a given interval.
We calculate definite integrals using the limits of integration. These are the two values defining the interval of interest. In our problem, we calculate the integral from 0 to infinity.
The basic process involves:

• Finding the antiderivative of the function.
• Evaluating this antiderivative at the upper limit and at the lower limit.
• Subtracting these two values to find the integral's result.

By performing these steps, you effectively find the 'exact' total area under the curve within the specified limits.

Exponential Functions

Exponential functions are a class of mathematical functions of the form \( f(x) = a e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. In this exercise, the function \( e^{-3x} \) is an exponential function, where \( a = 1 \) and \( b = -3 \).
Exponential functions are characterized by their constant relative growth rate, making them unique. They are used in many real-world applications, such as calculating compound interest and modeling population growth.
When integrating exponential functions, the result is typically another exponential function. However, if the function is bound by limits, as in definite integrals, we find a specific value. The antiderivative of \( e^{-3x} \) is \( -\frac{1}{3} e^{-3x} \), derived using the chain rule.
Due to the nature of exponential decay, functions like \( e^{-3x} \) trend toward zero as \( x \) tends to infinity, simplifying the calculation of improper integrals by ensuring a finite result, even when one of the limits is infinite.

Improper Integrals

Improper integrals extend the concept of definite integrals to deal with functions that have infinite limits of integration or unbounded behavior. These integrals offer a powerful tool for evaluating areas under curves stretching towards infinity.
The function \( \int_{0}^{\infty} c e^{-3x} \, dx \) is an example of an improper integral. Here, the upper limit of integration is infinity. To find this integral, we assume that the function behaves suitably and that its improper nature does not produce an undefined or divergent result.
To evaluate an improper integral:

• First, find the antiderivative of the function as usual.
• Then, consider the limit of this antiderivative as the upper bound approaches infinity.
• Combine this with the evaluation at the lower bound to find the result.

In this exercise, the improper nature simplifies because the exponential decay function rapidly approaches zero, ensuring a convergent solution. Thus, despite the infinite boundary, we can find a meaningful, finite value by evaluating this integral.