Areas Formed by Function Contours

To find the area between intersecting curves, we first identify their points of intersection, forming the boundaries of the area to calculate. By integrating the difference between the functions representing these curves over the defined interval, we can accurately determine the size of the area enclosed between them, whether straightforward or irregular in shape.

Calculating the area between curves mathematically involves the following steps:

1. Identify the Functions: Let’s have two functions, \( f(x) \) and \( g(x) \), representing the curves.
2. Find Intersection Points: Solve \( f(x) = g(x) \) to find the points of intersection. These points will serve as the limits of integration, let’s call them \( a \) and \( b \).
3. Set Up the Integral: The area \( A \) between the curves from \( x = a \) to \( x = b \) is given by the integral of the upper function minus the lower function. The integral is set up as:
   \( A = \int_{a}^{b} |f(x) – g(x)| \, dx \)
4. Evaluate the Integral: Calculate the integral. This step might require splitting the integral into multiple parts if the upper and lower functions change over the interval.
5. Calculate the Area: The result of the integral gives the total area between the two curves from \( x = a \) to \( x = b \).

The mathematical process essentially sums up the vertical slices of area between the curves, taking the absolute value to ensure the area is counted positively, regardless of which curve is on top.

Here is an example:
Calculate the area between the curve \( y = \sqrt{6x} \) and the x-axis, from \( x = 1 \) to \( x = 5 \).

Step-by-Step Solution:

1. Identify the Functions and Interval:

• Upper function: \( y = \sqrt{6x} \)
• Lower function: \( y = 0 \) (x-axis)
• Interval: from \( x = 1 \) to \( x = 5 \)

2. Set Up the Integral:

• Since the lower function is the x-axis, the area is simply the integral of the upper function over the interval.
• The integral is: \( \int_{1}^{5} \sqrt{6x} \, dx \)

3. Calculate the Integral:

• Find the antiderivative of \( \sqrt{6x} \), which is \( \frac{2}{3} \cdot (6x)^{3/2} \).
• Evaluate this from \( x = 1 \) to \( x = 5 \).

4. Calculate the Area:

• Substitute \( x = 5 \) and \( x = 1 \) into the antiderivative and calculate the difference:
  \( \frac{2}{3} \cdot 6^{3/2} \cdot 5^{3/2} – \frac{2}{3} \cdot 6^{3/2} \cdot 1^{3/2} \)

The result of this calculation gives the total area between the curve \( y = \sqrt{6x} \) and the x-axis from \( x = 1 \) to \( x = 5 \). The area is approximately \(16.624 \) square units.