Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A beam of light is shining onto a screen creating a disk of radius 50 centimeters. The intensity of light is brightest at the center and diminishes away from the center. If the intensity of light at a distance \(r\) from the center of the beam is given by \(f(r)=\frac{150}{20+r^{2}}\) watt/square \(\mathrm{cm}\), find the total wattage of the beam's image on the screen.

Step-by-Step Solution

Answer

The total wattage of the light beam on the screen is given by the value of the definite integral \(\int_{0}^{50}2\pi r f(r) dr\), with \(f(r)=\frac{150}{20+r^{2}}.\)

Step 1: Define the integral

We first define the integral that gives the total intensity of light. In this case, the integral is of the form \(\int_{0}^{50}2\pi r f(r) dr\), with \(f(r)=\frac{150}{20+r^{2}}.\)

Step 2: Carry out the integration

We now carry out the integration. This involves performing a basic antiderivative calculation, followed by evaluating the antiderivative at the bounds of integration (0 and 50).

Step 3: Evaluate the integral

After carrying out the integration, we find the definite integral by subtracting the value of the antiderivative at the lower limit from the value at the upper limit. This will give the total wattage.

Key Concepts

Integration

Integration is a central concept in calculus that helps us find the total or accumulated quantity across a range. In the context of the given problem, integration allows us to sum up the varying intensity of light over the entire area of the disk.
Using the integral, we can compute the total light intensity by adding up small, infinitesimal contributions of light at different distances from the center.

• It involves finding the integral of a function, which represents an infinite sum of the function's values over a specified interval.
• For our problem, we're using polar coordinates, which is why we see the term \(2\pi r\) in the integral. This accounts for the circular symmetry of the beam.

Understanding integration allows you to solve real-world problems that involve finding areas, volumes, and other quantities that change continuously.

Definite Integral

The definite integral is a specific type of integration that calculates the net area under a curve between two specific points, called the bounds of integration. In our light intensity problem, the definite integral \(\int_{0}^{50}2\pi r \frac{150}{20+r^{2}} dr\) calculates the total wattage of the beam as it shines onto the screen.

• The bounds (0 and 50) represent the limits from the center of the beam to its edge.
• The definite integral evaluates to an exact numerical value, representing the accumulated light intensity across the disk.

By subtracting the antiderivative evaluated at the lower bound from that at the upper bound, you arrive at the total intensity—your solution to the problem!

Antiderivative

An antiderivative is essentially the reverse of a derivative. It is a function whose derivative yields the original function. In solving our exercise, we must find the antiderivative of \(f(r) = \frac{150}{20 + r^2}\) to determine how the light intensity accumulates over different radii.

• Finding the antiderivative helps transform the rate of change (intensity function) into a cumulative measure (total wattage).
• Once you have the antiderivative, you evaluate it at the specific boundaries (0 and 50 in this case), allowing you to perform the definite integral process.

Understanding antiderivatives is vital for solving integration problems where you want to find accumulated quantities like area, volume, or in this case, light intensity.

Light Intensity

Light intensity refers to the power per unit area that a light source emits. In our problem, the light's intensity is brightest at the center and decreases with distance from the center according to the function \(f(r) = \frac{150}{20 + r^2}\).

• The function model provides a mathematical representation of how brightness decreases as we move away from the center.
• By integrating this function across the disk, we're calculating the total wattage or energy output of the beam.

Understanding how light intensity functions help in numerous applications, from designing lighting systems to analyzing radiation patterns. It gives a comprehensive picture of the behavior and distribution of light in different environments.