Problem 28 - Applied Calculus

Question

(a) Use a graph of the integrand to make a rough estimate of the integral. Explain your reasoning. (b) Use a computer or calculator to find the value of the definite integral. $$\int_{0}^{1} 3^{t} d t$$

Step-by-Step Solution

Answer

Estimate: 1 to 3, Exact: ~2.186.

Step 1: Understanding the Integrand

The integrand given is the function \( 3^t \). Since it is an exponential function with base greater than 1, we know that it will be increasing as \( t \) increases from 0 to 1.

Step 2: Sketching the Graph

Sketch the graph of \( 3^t \) over the interval [0, 1]. At \( t = 0 \), \( 3^0 = 1 \) and at \( t = 1 \), \( 3^1 = 3 \). This means the graph starts at 1 and rises smoothly to 3. The graph is an increasing curve.

Step 3: Estimating the Area Under the Curve

The area under the curve from \( t = 0 \) to \( t = 1 \) represents the integral. Since the graph is above the line \( y = 1 \) and below \( y = 3 \), a rough estimate of the integral's value is between 1 and 3. We can improve our estimate by considering the curve's upward slope is more rapid near \( t = 1 \). Thus, a reasonable estimate could be closer to 2.

Step 4: Calculating Using Technology

Use a computer or calculator to find the precise value of the integral: \( \int_{0}^{1} 3^t \, dt \). Inputting this into a calculation tool gives approximately 2.186.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where the variable is in the exponent. In this case, our function is \( 3^t \). The distinctive trait of an exponential function is its rapid growth. As the base (here, 3) is greater than 1, the function will increase quickly as you move from left to right along the x-axis. For exponential functions:

• The graph will be a smooth curve.
• If the base is greater than 1, like 3, the function grows.
• As \( t \) increases, \( 3^t \) becomes significantly larger.

This property makes exponential functions critical in studying phenomena involving growth and decay, such as population growth or radioactive decay. In our exercise, when \( t \) moves from 0 to 1, \( 3^t \) transitions from 1 to 3, illustrating this rapid growth. This understanding is crucial for sketching the graph accurately, which is our first step in visualizing the integral.

Area Under the Curve

The definite integral of a function over an interval, such as \( \int_{0}^{1} 3^t \, dt \), represents the area under the function’s curve between two points. In simple terms, it sums up infinitely small rectangles into which the area is divided.For the exponential function \( 3^t \):

• We are interested in the interval from \( t = 0 \) to \( t = 1 \).
• The curve starts at \( y = 1 \) and rises to \( y = 3 \).
• The area under the curve will fall between these two y-values.

An important detail here is that since \( 3^t \) is an increasing function, the greater height near \( t = 1 \) means that the area under the curve is weighted more heavily towards the higher value of 3. Estimating this area provides insight into the behavior of the integral. In this case, it's reasonable to anticipate that the area under \( 3^t \), which captures the integral, will fall close to 2. This observation aligns with our intuitive need to account for the rapid increase of the function.

Graphing Integrals

Graphing integrals starts with sketching the function described by the integrand. For our function \( 3^t \), the graph forms a curve that helps visualize the area under it. Graphing is invaluable in providing a visual interpretation of the function, making it easier to estimate integrals.When graphing \( 3^t \):

• At \( t = 0 \), the value is 1, and at \( t = 1 \), it equals 3, showing a clear upward trend.
• By plotting these points and smoothly connecting them, you can see how the area accumulates under the curve.
• It helps visually bound the integral's area between our two horizontal lines, \( y = 1 \) and \( y = 3 \).

This visual approach effectively informs us that \( \int_{0}^{1} 3^t \, dt \) is more than just a number; it's an infinitesimal sum that manifests as the concept of area. By using computational tools post-graphing, we not only confirm our rough estimates but also appreciate the intricacies of the curve. Hence, combining graphing and technology solidifies understanding both conceptually and quantitatively.