Problem 32 - Calculus of a Single Variable

Question

Finding the Area of a Region In Exercises \(31-36,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region analytically, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x^{4}-2 x^{2}, \quad y=2 x^{2} $$

Step-by-Step Solution

Answer

The area of the region bounded by given graphs is found using definite integrals.

Step 1: Graph the Equations

Graph the equations \(y=x^{4}-2 x^{2}\) and \(y=2 x^{2}\) using a graphing utility. Identify the points where the graphs intersect. These will be the limits of integration.

Step 2: Find Intersection Points

To find the intersection points, set the two equations equal to each other and solve for x: \(x^{4}-2 x^{2}=2 x^{2}.\) This simplifies to \(x^{4}-4x^{2}=0\). Factoring gives \(x^{2}(x^{2}-4)=0\). Which means x equals 0, -2, or 2.

Step 3: Set up the Integral

Now, set up the definite integral to find the area between the two curves. Remember that the area is the integral of the top function minus the bottom function, evaluated between the limits of integration. Here it is: \(\int_{-2}^{2}(2x^{2}) dx - \int_{-2}^{2}(x^{4}-2 x^{2}) dx\)

Step 4: Evaluate the Integral

Evaluate the integrals, apply the fundamental theorem of calculus and simplify. This gives the area of the bounded region.

Step 5: Verify Results using Graphing Utility

Now use the integration capabilities of the graphing utility to check the results obtained in the previous step.

Key Concepts

Intersection Points

Understanding intersection points is essential when finding the area between two curves. These are the points where the graphs of the equations meet or cross each other. To solve for intersection points analytically, you set the two functions equal to each other and solve for the variable. In the exercise, we need to find the intersection points of the equations:

• \(y = x^4 - 2x^2\)
• \(y = 2x^2\)

To find these points, we equate both equations: \(x^4 - 2x^2 = 2x^2\). Simplifying, this becomes \(x^4 - 4x^2 = 0\). Factor it to \(x^2(x^2 - 4) = 0\). Therefore, the solutions are \(x = 0\), \(x = -2\), and \(x = 2\). These solutions are key because they represent the x-values where the graphs intersect and will serve as limits for our integration, determining the area between the curves.

Definite Integral

The definite integral is used to calculate the exact area between two curves. Essentially, it sums up all the infinitesimally small segments of area from one boundary to another. For our exercise, the definite integral allows us to find the area between the curves \(y = x^4 - 2x^2\) and \(y = 2x^2\) over the interval, determined by the intersection points \(-2\) to \(2\). When setting up the integral for the area between these curves, compute the integrals of the top function \(2x^2\) and bottom function \(x^4 - 2x^2\) separately, and then subtract the second integral from the first:

• Top function: \(\int_{-2}^{2} 2x^2 \, dx\)
• Bottom function: \(\int_{-2}^{2} (x^4 - 2x^2) \, dx\)

Subtracting these integrals \(\int_{-2}^{2} 2x^2 \, dx - \int_{-2}^{2} (x^4 - 2x^2) \, dx\) gives the area between the curves.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges differentiation and integration, providing a practical way to evaluate definite integrals. It tells us that if we have a continuous function, the definite integral over the interval from \(a\) to \(b\) can be calculated by finding the anti-derivative of the function, then subtracting the value of this anti-derivative at \(a\) from its value at \(b\). In applying the Fundamental Theorem of Calculus to our problem: We compute the integral for \(\int_{-2}^{2} 2x^2 \, dx\) and \(\int_{-2}^{2} (x^4 - 2x^2) \, dx\) separately. After finding the antiderivatives, evaluate them at \(x = 2\) and \(x = -2\), then subtract. This process yields the total area between the curves, providing a straightforward numerical result that helps verify our analytical solution.

Graphing Utility

A graphing utility is a powerful tool that helps visualize equations and verify calculations. In our exercise, the graphing utility assists in several ways, making the process of finding the area between curves both visual and precise. Initially, you can use a graphing utility to plot the functions \(y = x^4 - 2x^2\) and \(y = 2x^2\). This visual representation helps in identifying the intersection points, which are essential for determining our limits of integration. Furthermore, graphing calculators or software typically have features that allow you to compute the definite integral directly. By using the integration capabilities of a graphing utility, you can verify the results you calculated analytically. This double-check ensures accuracy and builds confidence in your mathematical operations.