Question

Find the total area enclosed by the functions \(f\) and \(g\). \(f(x)=x^{3}-4 x^{2}+x-1, g(x)=-x^{2}+2 x-4\)

Short answer

The total area is found by integrating the difference of functions between their intersections.

Step-by-step solution

Find the Points of Intersection

To find the points of intersection between the two functions, set them equal to each other: \[x^3 - 4x^2 + x - 1 = -x^2 + 2x - 4\]Rearrange the equation:\[x^3 - 4x^2 + x - 1 + x^2 - 2x + 4 = 0\]Combine like terms:\[x^3 - 3x^2 - x + 3 = 0\]Now, solve this cubic equation for the values of \(x\). This can involve methods like factoring, synthetic division, or using a graphing calculator to approximate the roots.

Solve for the Intersection Points

A graphing calculator or algebraic tools can be used to find the roots of the polynomial:\[x = 3, \, x \approx 1.13, \, x \approx -0.13\] These are the points where the two curves intersect: \(x = 3\), \(x \approx 1.13\), and \(x \approx -0.13\).

Set Up the Integral to Find the Area

The function that is 'on top' usually bounds the area to the specified points. We exert our attention primarily on differencing expressions \( f(x) \) (on top) and \( g(x) \) (bottom) within the intersection points:\[ \text{Area} = \int_{-0.13}^{1.13} ((x^3 - 4x^2 + x - 1) - (-x^2 + 2x - 4)) \, dx + \int_{1.13}^{3} ((-x^2 + 2x - 4) - (x^3 - 4x^2 + x - 1)) \, dx \]

Key concepts

Points of intersection

To find where two curves meet, you need to determine their points of intersection. This is done by setting their equations equal to each other. Given the functions \(f(x) = x^{3} - 4x^{2} + x - 1\) and \(g(x) = -x^{2} + 2x - 4\), the first step is to equate them: \[x^3 - 4x^2 + x - 1 = -x^2 + 2x - 4\]This equation helps find the x-values where the curves cross. To simplify and solve, you rearrange terms: \[x^3 - 4x^2 + x - 1 + x^2 - 2x + 4 = 0\] By combining like terms, we get: \[x^3 - 3x^2 - x + 3 = 0\]. This tells us the x-coordinates where intersections happen, helping us determine where to focus further calculations.

Integral setup

Once you have the points of intersection, setting up integrals helps find the area between the curves. For the functions in question, their intersection points were determined as \(x = 3\), \(x \approx 1.13\), and \(x \approx -0.13\). Determining which curve is on top within each interval is essential, as calculating area involves subtracting the lower curve from the upper curve.The integral for the area between the functions \(f(x)\) and \(g(x)\) from \(-0.13\) to \(1.13\) is: \[\int_{-0.13}^{1.13} ((x^3 - 4x^2 + x - 1) - (-x^2 + 2x - 4)) \, dx\]This segment relies on \(f(x)\) being above \(g(x)\).From \(1.13\) to \(3\), \(g(x)\) outweighs \(f(x)\), so the integral setup changes to: \[\int_{1.13}^{3} ((-x^2 + 2x - 4) - (x^3 - 4x^2 + x - 1)) \, dx\]In essence, this step ensures the calculation of the respective areas in the right order.

Cubic equation

The task of solving the cubic equation \(x^3 - 3x^2 - x + 3 = 0\) is crucial for determining the intersection points. Cubic equations, like this one, have three roots—these could be real or complex numbers. However, in scenarios involving curves, we're interested in real roots that indicate the intersection points on the graph. A cubic equation can be approached through different methods:

• Factoring: Although this isn't always simple with cubics, factoring can help if the equation can be neatly decomposed into a product of simpler polynomials.
• Graphing Calculators: These tools can estimate roots effectively by visualizing where the curve crosses the x-axis.
• Synthetic Division: Used if a possible rational root is suspected to confirm it's a solution.

In this specific problem, using a graphing calculator or software can quickly identify the real roots needed for the intersection points.

Polynomial roots

Understanding polynomial roots is essential for discovering where functions intersect. In the given exercise, solving the polynomial \(x^3 - 3x^2 - x + 3 = 0\) furnishes the necessary intersection points for calculating area. To solve for roots, various strategies can be employed:

• Rational Root Theorem: This suggests possible rational solutions which can be verified through substitution.
• Graphical Analysis: Using digital tools to plot the polynomial helps visualize roots, simplifying identification.
• Trial and Error: Substituting guessed values to see if they satisfy the equation.

Roots such as \(x = 3\), \(x \approx 1.13\), and \(x \approx -0.13\) convey where the two functions meet, enabling the calculation of the area between them via integrals.