Problem 54 - Applied Calculus

Question

Find the area bounded by the given curves. \(y=x^{3}\) and \(y=x^{4}\)

Step-by-Step Solution

Answer

The area is \(\frac{1}{20}\).

Step 1: Sketch the curves

Visualize the given curves by sketching them or by using a graphing tool. The curve \(y = x^3\) is a cubic function, while \(y = x^4\) is a quartic function. Notice that \(y = x^4\) is closer to the x-axis for \(|x| < 1\) because it grows slower than \(y = x^3\) in this interval.

Step 2: Find intersection points

Set \(y = x^3\) equal to \(y = x^4\) to find the intersection points: \(x^3 = x^4\). Solve for \(x\) by factoring: \(x^3 (1 - x) = 0\). Therefore, the intersection points are \(x = 0\) and \(x = 1\).

Step 3: Set up the integral for the area

The area between the two curves from the intersection points involves integrating the difference of the functions. The area \(A\) is given by: \[ A = \int_0^1 (x^3 - x^4) \, dx \] where \(x^3\) is greater than \(x^4\) over the interval \([0, 1]\).

Step 4: Integrate the function

Compute the definite integral:\[ \int_0^1 (x^3 - x^4) \, dx = \left[ \frac{x^4}{4} - \frac{x^5}{5} \right]_0^1 \] Evaluating this gives: \[ = \left( \frac{1}{4} - \frac{1}{5} \right) - (0) = \frac{1}{4} - \frac{1}{5} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \]

Step 5: Conclusion and interpretation

The calculated area represents the region bounded by the curves \(y = x^3\) and \(y = x^4\) from \(x = 0\) to \(x = 1\).

Key Concepts

Cubic function

A cubic function is a polynomial of degree three, with the general form \( y = ax^3 + bx^2 + cx + d \). This type of function is known for its characteristic S-shaped curve, which can exhibit one or two bends, depending on the nature of its coefficients. The simplest form of a cubic function is \( y = x^3 \), which is strictly increasing and symmetrical with respect to the origin. This symmetry means that as \( x \) becomes more negative, \( y \) decreases, and as \( x \) becomes more positive, \( y \) increases.

In the exercise, the curve \( y = x^3 \) is used, which is a standard cubic function. Its steep slope near the origin becomes more pronounced as \( x \) moves away from zero. Understanding how cubic functions behave, especially around crucial points like roots and inflection points, helps to accurately analyze their intersection with other functions, such as the quartic function used in this problem.

Key characteristics of cubic functions include:

• They can have up to three real roots.
• Their graphs can feature one or two turning points.
• They exhibit odd symmetry, which means they are not symmetrical across any axis but rather through rotation by 180° about their central point.

If you encounter a graph involving a cubic function in an area problem, identify these features to understand its contribution to the bounded region.

Quartic function

A quartic function is a polynomial expression of degree four, typically presented in the form \( y = ax^4 + bx^3 + cx^2 + dx + e \). These functions are noteworthy because of their potentially complex graph shapes, which can have several turning points. The simplest form, \( y = x^4 \), is a parabolic shape that opens upwards and is symmetric about the y-axis.

In our scenario with \( y = x^4 \), the function is less steep near the origin compared to \( y = x^3 \), highlighting an essential aspect of quartic functions: they tend to grow more slowly than cubic ones for smaller values of \( |x| \). This slower growth affects the area and integration calculations when paired with cubic functions.

Characteristics of quartic functions include:

• They can have up to four real roots.
• Their graphs contain multiple turning points, potentially up to three.
• Their symmetry is even, as \( y = x^4 \) is symmetric around the y-axis.

When working with quartic functions, especially in area problems, being able to visualize their extended growth and comparative behavior to cubic functions is beneficial in understanding how they form the bounds of a region.

Definite integral

The definite integral is a fundamental concept in calculus used to find the area under a curve between two points. In essence, it calculates the accumulation of quantities, which can represent areas, volumes, displacement, etc. For the problem at hand, we're using it to determine the area between the functions \( y = x^3 \) and \( y = x^4 \).

The integral from \( a \) to \( b \) of a function \( f(x) \) is denoted as \( \int_a^b f(x) \, dx \). It evaluates how much area is under the curve of \( f(x) \) between the points \( x = a \) and \( x = b \). In the context of finding the area between two curves, you set up an integral with the difference of the functions, such as \( \int_a^b (f(x) - g(x)) \, dx \) where \( f(x) \) is above \( g(x) \) over the interval.

For instance, in our exercise:

• We found the boundaries of integration by solving \( x^3 = x^4 \).
• The area was calculated using \( \int_0^1 (x^3 - x^4) \, dx \).
• Computing this integral gives the precise area between \( x = 0 \) and \( x = 1 \).

Practicing how to set up and solve definite integrals is crucial in evaluating and interpreting areas between curves. Understanding these integrals will help illuminate how regions within coordinate planes correspond to numerical outputs of integrals.