Problem 49 - Basic Technical Mathematics with Calculus

Question

A coffee-table top is designed to be the region between \(y=0.25 x^{4}\) and \(y=12-0.25 x^{4}\) (dimensions in \(\mathrm{dm}\) ). What is the area of the table top?

Step-by-Step Solution

Answer

The area of the table top is 48 dm².

Step 1: Understand the Functions

The given functions describe the boundaries of the coffee table top, where the area lies between \(y = 0.25x^4\) (a quartic function opening upwards) and \(y = 12 - 0.25x^4\) (a quartic function opening downwards). These two curves form a symmetrical shape where the x-axis acts as the line of symmetry.

Step 2: Finding Points of Intersection

To find the limits for integration, set the functions equal to each other: \(0.25x^4 = 12 - 0.25x^4\). Simplify and solve for \(x\): \(0.5x^4 = 12\), so \(x^4 = 24\). Thus, \(x = \pm \sqrt[4]{24}\). Therefore, the points of intersection are \((-\sqrt[4]{24}, 6)\) and \((\sqrt[4]{24}, 6)\).

Step 3: Determine the Area using Integration

The area between the two curves can be calculated by integrating the top function minus the bottom function: \( \int_{-\sqrt[4]{24}}^{\sqrt[4]{24}} \left(12 - 0.25x^4 - 0.25x^4\right) \, dx \). Simplifying the integrand gives \( \int_{-\sqrt[4]{24}}^{\sqrt[4]{24}} (12 - 0.5x^4) \, dx \).

Step 4: Compute the Integral

Evaluate the integral: \( \int 12 \, dx - \int 0.5x^4 \, dx\). This becomes \([12x - 0.1x^5]_{-\sqrt[4]{24}}^{\sqrt[4]{24}}\). Substitute the limits of integration to find the area: \[ \left[12(\sqrt[4]{24}) - 0.1(\sqrt[4]{24})^5\right] - \left[12(-\sqrt[4]{24}) - 0.1(-\sqrt[4]{24})^5\right]. \]

Step 5: Simplify and Calculate the Result

On plugging the values:1. The expression becomes \(2 \times \left[12 \times \sqrt[4]{24} - 0.1 \times (24 \times \sqrt[4]{24})\right]\).2. Calculate \(2 \times 12 \times \sqrt[4]{24}\) and \(2 \times 0.1 \times 24 \times \sqrt[4]{24}\).3. Simplify and calculate the numerical result.The computed area of the coffee table top is \(48\) dm².

Key Concepts

Quartic Functions

Quartic functions are polynomial functions of degree four. These functions are of the form \(y = ax^4 + bx^3 + cx^2 + dx + e\). In the case of our exercise, the quartic functions are simplified into two equations: \(y = 0.25x^4\) and \(y = 12 - 0.25x^4\). The first opens upwards, forming a U-shape, while the second opens downwards, forming an inverted U. Together, they create a symmetrical area where each curve acts as a boundary.

In general, the shape of the graph is determined by the leading coefficient (the term with \(x^4\)). When it is positive, the graph opens upwards, and when it is negative, it opens downwards. Understanding the orientation and position of quartic functions is crucial when calculating areas or finding intersections. This knowledge helps in visualizing how curves interact with one another.

Area Between Curves

The concept of finding the area between curves is essential in calculus integration. This involves computing the region enclosed by two or more functions. To find this area, you subtract the lower function from the upper function and integrate over the interval defined by points of intersection.

In our example, the area of the coffee table surface is the region between \(y = 12 - 0.25x^4\) (upper function) and \(y = 0.25x^4\) (lower function). The formula used is:

• \( \int (\text{upper function} - \text{lower function}) \, dx \)

This method ensures that you calculate the exact area contained within the boundaries of the two quartic functions, preserving their geometric relationship.

Points of Intersection

Points of intersection are critical when determining the area between curves. These points occur where the functions have the same value, i.e., they intersect on the graph.

To find them, set the expressions for the two functions equal: \(0.25x^4 = 12 - 0.25x^4\). Solving this yields \(x^4 = 24\), and hence \(x = \pm \sqrt[4]{24}\). This results in the intersection points \((\pm \sqrt[4]{24}, 6)\).
The intersection points define the limits of the region to be integrated. They tell you precisely where the area of interest starts and ends along the x-axis.

Integration Limits

Integration limits are vital as they define the span of the x-axis over which you will calculate the area between the curves. These limits are directly obtained from the points of intersection.

For this exercise, the integration limits are determined from \(x = -\sqrt[4]{24}\) to \(x = \sqrt[4]{24}\). This symmetrical range reflects the design of the coffee table top and shows the extent of our integration.
Once you have the limits, you can set up the integral. The range ensures you're calculating the full area contained within the curves between these two points. It guides the evaluation of the integral to provide a complete and accurate measurement of the area.