Problem 41 - Algebra 2

Question

Open-Ended Write equations for three curves that are positive for \(1 \leq x \leq 3\) . Use your graphing calculator to find the area under each curve for this domain.

Step-by-Step Solution

Answer

The area under the curve for the respective equations \(f(x) = x + 1\), \(g(x) = (x-2)^2 + 1\), and \(h(x) = (x-2)^3 + 3\) can be found by performing definite integration from 1 to 3 for each function using a graphing calculator. The definite integral values provided by the calculator represent the areas under each curve.

Step 1: Formulation of Positive Function Equations

Let's start by formulating three positive functions i.e., linear, quadratic, and cubic functions in the given range. A linear function can take the form \(f(x) = x+1\) which is positive for \(x \geq 1\). A simple quadratic function with positive values can be \(g(x) = (x-2)^2 + 1\) and a cubic function can be \(h(x) = (x-2)^3 + 3\).

Step 2: Calculating Area under Each Curve via Integration

The area under each curve from 1 to 3 can be found using definite integration. The area under the curve \(f(x)\) can be calculated as \(\int_{1}^{3} (x+1) dx\). Similarly, the area under \(g(x)\) is given by \(\int_{1}^{3} (x-2)^2 + 1 dx\), and \(h(x)\) area is given by \(\int_{1}^{3} (x-2)^3 + 3 dx\).

Step 3: Use of the Graphing Calculator for Integration

Now, using a graphing calculator, enter each function and compute the integrals between 1 and 3. The calculator should provide definite integrals which represent the area under each curve in the defined range.

Key Concepts

Definite Integration

In integral calculus, definite integration is a process used to calculate the area under a curve between two specific points on the x-axis. When given a function, the definite integral provides the accumulated total between these two bounds. The result is a number, which represents the net area in terms of the units used. This differs from an indefinite integral, which represents a family of functions and includes an arbitrary constant (usually noted as +C).

For instance, if you have a function like \(f(x) = x + 1\), and you'd like to find the area from 1 to 3, you perform the definite integration \[ \int_{1}^{3} (x + 1) \, dx. \]Here, you apply the fundamental theorem of calculus, which requires evaluating the antiderivative of the function at both bounds and subtracting:

• Find the antiderivative of \(f(x) = x + 1\), which is \(F(x) = \frac{x^2}{2} + x\).
• Evaluate \(F(x)\) at 3 and 1, then subtract: \(\left(\frac{3^2}{2} + 3\right) - \left(\frac{1^2}{2} + 1\right)\).

This gives the net area under the curve from 1 to 3. Definite integration is crucial for finding areas, solving physics problems, and calculating probabilities.

Positive Functions

Positive functions are functions whose outputs are always greater than or equal to zero over a specified interval. In the context of integral calculus, a positive function is useful when determining the area under the curve, as the area will directly correspond to the definite integral value without requiring additional adjustments for negative results.

When defining positive functions, especially between a range, it is important to ensure the function does not dip below the x-axis. With the exercise at hand:

• Understanding simple linear functions like \(f(x) = x + 1\), which stay positive for \(x \geq 1\) since adding 1 keeps it above zero.
• Exploring quadratic functions such as \(g(x) = (x-2)^2 + 1\) where the square component ensures non-negativity, and adding 1 shifts the entire curve upwards.
• Considering cubic functions like \(h(x) = (x-2)^3 + 3\) that are adjusted up by 3, ensuring positivity in the specified range.

Using positive functions helps simplify the process of calculating areas under curves because the results from definite integration perfectly reflect those areas without requiring modifications to account for portions of graphs beneath the x-axis.

Graphing Calculator Usage

A graphing calculator is an invaluable tool for visualizing functions and performing complex calculations such as integrals. This technology assists students in verifying their work and understanding the graphical representation of mathematical concepts.

When using a graphing calculator for definite integration:

• Input the function. For example, enter \(f(x) = x + 1\) into the calculator.
• Use the calculator's integration function, often found in the "math" menu under an option like "fnInt" or an integration symbol, to integrate from 1 to 3.
• Ensure the calculator is set to the correct mode (e.g., "theme" to decimal) for precise results.

The calculator simplifies the integration process by automatically computing the integral and outputting the exact area between the specified bounds. Visual graphs provide additional understanding by showing how the area looks beneath the plotted curve. This dual approach of numeric and visual feedback supports a deeper comprehension of integral calculus principles.