Problem 25 - Calculus: Early Transcendentals

Question

Sketch the region enclosed by the given curves and find its area. \( y = x^4 \) , \( y = 2 - \mid x \mid \)

Step-by-Step Solution

Answer

Determine intersection points, sketch the curves, and integrate their difference over the domain.

Step 1: Identify Points of Intersection

To find the region enclosed by the curves, we first need to identify the points where they intersect. Set the equations equal to each other: \[ x^4 = 2 - |x| \]This equation needs to be solved for \(x\). Notice that \(|x|\) can be either \(x\) or \(-x\). Therefore, we consider both cases:

Step 1a: Solve for \(x \geq 0\)

For positive \(x\), \(|x| = x\), and our equation becomes:\[ x^4 = 2 - x \]Rearrange terms:\[ x^4 + x - 2 = 0 \]Find roots using numerical methods or a graphing calculator to determine intersections in this case.

Step 1b: Solve for \(x < 0\)

For negative \(x\), \(|x| = -x\), and our equation becomes:\[ x^4 = 2 + x \]Rearrange terms:\[ x^4 - x - 2 = 0 \]Similarly, find roots using numerical methods or a graphing calculator to determine intersections.

Step 2: Sketch the Curves

Sketch \(y = x^4\) which is a symmetric curve about the y-axis and opens upwards, and \(y = 2 - |x|\), a V-shape with a maximum at \(x = 0\) (y-intercept at 2), and decreases linearly. The point of intersection determined from step 1 will serve as the limits of the region.

Step 3: Setup the Integral for Area Calculation

The area of the region between two curves \(y = f(x)\) and \(y = g(x)\) over the interval \([a, b]\) is given by:\[ \int_{a}^{b} (g(x) - f(x))\,dx \]Here, \(g(x) = 2 - |x|\) and \(f(x) = x^4\). Identify the interval based on the intersection points determined.

Step 4: Evaluate the Definite Integral

Evaluate \[ \int_{-a}^{a} ((2 - |x|) - x^4) \, dx \]The interval limits a need to be confirmed from the intersection points found earlier. Note that symmetry can be used to simplify calculations by evaluating from 0 to a and doubling the result if the area is symmetric.

Step 4a: Calculate Area Contribution for x>=0

For \(x \geq 0\), the integral becomes:\[ \int_{0}^{a} (2 - x - x^4) \, dx \]Integrate each term separately and compute the definite integral values using the limits.

Step 4b: Calculate Area Contribution for x<0

For \(x < 0\), the integral can be evaluated as:\[ \int_{-a}^{0} (2 + x - x^4) \, dx \]Similarly, integrate each term, apply limits and calculate the area contribution.

Step 5: Combine Area Contributions

Add the area contributions from both halves of the region (or multiply by 2 if symmetric of evaluated integrals from Step 4a or 4b). The result is the total enclosed area between the curves.

Key Concepts

Intersection Points

Finding intersection points is crucial when determining the area between curves. These points are where two curves meet, indicating the boundaries of the region you are interested in.
For the curves given, namely, \( y = x^4 \) and \( y = 2 - |x| \), we must set them equal to find these points.
That gives the equation \( x^4 = 2 - |x| \), which needs solving for \(x\).
Considering the absolute value \(|x|\):

• When \(x \geq 0\), \(|x| = x\) and the equation simplifies to \( x^4 = 2 - x \).
• For \(x < 0\), \(|x| = -x\) and it becomes \( x^4 = 2 + x \).

Each case requires solving the resulting polynomial equation, which can be approached using numerical methods, graphing calculators, or software tools.
The solutions give us the \(x\) values of the intersections, thus defining the integral bounds needed for area calculation.

Definite Integral

The definite integral allows us to calculate the exact area between two curves over a specific interval. This is essential for finding the enclosed region between \( y = x^4 \) and \( y = 2 - |x| \).
The fundamental principle is:

• The area between the curves \( y = g(x) \) and \( y = f(x) \) from \(x = a\) to \(x = b\) is given by \( \int_{a}^{b} (g(x) - f(x)) \, dx \).

In our case:

• \(g(x) = 2 - |x|\) and \(f(x) = x^4\).
• The limits \([a, b]\) are determined from the intersection points.
• For a symmetric region, calculations can be simplified. When the region is symmetric about an axis, you might only need to calculate half of the area and double it for the full region.

Solving these integrals involves calculating each component of the integral, evaluating it at the intersection points, and subtracting where necessary.

Curve Sketching

Sketching curves is a valuable technique that provides a visual representation, making it easier to understand the regions involved. For the functions \( y = x^4 \) and \( y = 2 - |x| \), you can follow these guidelines:

• \( y = x^4 \) is a symmetric curve about the y-axis, opening upwards with its vertex at the origin (0,0).
• \( y = 2 - |x| \) forms a V-shape, peaking at \( (0, 2) \) and sloping downwards. It is symmetric about the y-axis, with lines from \( (0, 2) \) to \((2, 0)\) and \((-2, 0)\).

By sketching both curves, you can identify early on where they intersect, allowing you to better understand their relationship and the enclosed area to calculate.

Symmetric Region

Symmetry in a region can greatly simplify the process of finding the area between curves. The symmetry implies that the shape is mirrored about a certain line, usually an axis.
This is highly useful, as shown in the problem with \( y = x^4 \) and \( y = 2 - |x| \), where the region is symmetric about the y-axis.
Some key points to note:

• Symmetry means that computing the area for half the region (say, from \( x = 0 \) to an endpoint \( x = a \)) can be doubled to find the total area.
• When dealing with symmetry, it can lead to simpler integrations as only one side of the curve is considered.

Utilizing symmetry not only saves time in calculations but also reduces the risk of error by minimizing repetitive computations.