Question

Find the area of the propeller-shaped region enclosed between the graphs of $$y=\frac{2 x}{x^{2}+1} \quad$$ and $$\quad y=x^{3}$$

Short answer

The area of the propeller-shaped region enclosed between the given graphs is \(4-\frac{2}{3}\) square units.

Step-by-step solution

Sketch the graphs

Sketch the graphs of both functions \(y=\frac{2x}{x^{2}+1}\) and \(y=x^{3}\) to visualize the propeller-shaped region. Using graphing software or an accurate hand sketch will be beneficial.

Find the points of intersection

The points of intersection are found by setting the two equations equal and solving for \(x\). This leads us to:\[\frac{2x}{x^{2}+1}=x^{3}\]Solving this equation we find the intersection points at \(x=-1, 0, 1\).

Determine the integral

To find the area between two curves, subtract the lower function from the higher function, then integrate according to the limits. Here,\[\int_{-1}^{1}[(\frac{2x}{x^{2}+1})-(x^{3})]\,dx\] is the integral to solve. Note that on the interval from -1 to 1, \(\frac{2x}{x^{2}+1}\) is the bigger function.

Solve the integral

When we simplify and solve the above integral, we get: \[\int_{-1}^{1}(\frac{2x}{x^{2}+1}-x^{3})\,dx=4-\frac{2}{3}\]

Compute the result

Now, subtracting elevation of the \(x^{3}\) function from the elevation given by the function \(\frac{2x}{x^{2}+1}\) gives the final result.

Key concepts

Integrating Functions

Integrating functions is a fundamental concept in calculus that allows us to calculate the area under a curve, among other things. When we have two functions, say function 'A' and function 'B', we can find the area between them on a certain interval by integrating the difference between them. This is precisely what we do when we want to find the area between two curves. We establish the limits of integration, which are the x-values that frame the area we're interested in, and during the integration process, we subtract the value of the lower function from the value of the higher function at every point along the interval. The definite integral – indicated by integral signs with upper and lower limits – provides a way to calculate cumulative values, such as the area, by effectively 'adding up' an infinite number of infinitesimally small quantities.

Finding Points of Intersection

Finding the points of intersection between two functions is critical in understanding where their graphs intersect, which in turn is essential for determining the area between the functions. To find these points, we set the equations of the two functions equal to each other and solve for the variable, typically 'x'. This process might involve simplifying expressions, factoring, or applying algebraic methods such as the quadratic formula. It's important to keep in mind that not all intersection points are created equal: the context of the problem may require us to only consider intersections within a certain range or domain.

Graphing Functions

Graphing functions is a visual representation of the relationship between two variables, usually 'x' and 'y'. By plotting a function on a coordinate plane, we can see its shape and behavior, such as where it increases, decreases, or has any asymptotes or discontinuities. Graphing is not only a powerful tool for understanding the behavior of functions, but it is also quintessential for problems involving areas between curves, as it helps us visualize where one function lies above or below another and to identify the relevant intervals for integration. In an educational setting, it's often suggested to use graphing calculators or software for a more precise representation.

Definite Integral Application

The application of the definite integral in calculus to find the area between curves is a practical and frequent use of integration. Once the functions are defined, the points of intersection are found, and the graph is understood, setting up the integral is rather straightforward. The integral itself consolidates the infinite sum of products of the differences of function values and an infinitesimally small change in 'x' over the chosen interval. This calculates the 'net' area between curves – accounting for regions where one curve might be below another. The power of the definite integral application lies in its ability to summarize continuous and often complex relationships into a single, comprehendible value, which in many cases represents a physical quantity such as area.