Area of Shaded Region Integral Calculator

What is the Area of a Shaded Region?

The area of a shaded region between two curves is the space enclosed by those curves over a specific interval. In calculus, this area is calculated using definite integrals, which sum up the area of infinitesimally thin vertical strips between the curves.

This concept is widely used in physics to find work done by variable forces, in engineering to calculate areas of complex shapes, and in economics to determine the area under a demand curve.

How to Calculate the Area of a Shaded Region

To calculate the area between two curves using integrals, follow these steps:

1. Identify the upper and lower functions that bound the shaded region.
2. Determine the interval [a, b] over which you're calculating the area.
3. Set up the integral as the difference between the upper and lower functions.
4. Evaluate the definite integral to find the exact area.

This process is particularly useful when the area cannot be easily calculated using geometric formulas.

The Formula

The area A between two curves y = f(x) (upper function) and y = g(x) (lower function) from x = a to x = b is given by:

Where:

• f(x) is the upper function
• g(x) is the lower function
• [a, b] is the interval of integration

Worked Example

Let's calculate the area between the curves y = x² and y = x from x = 0 to x = 2.

1. Identify the upper and lower functions: f(x) = x² (upper), g(x) = x (lower).
2. Set up the integral: ∫[0 to 2] (x² - x) dx.
3. Evaluate the integral:
   ∫(x² - x) dx = (x³/3 - x²/2) evaluated from 0 to 2 = [(8/3 - 4/2) - (0 - 0)] = [8/3 - 8/2] = [16/6 - 24/6] = -8/6 = -4/3
4. The negative sign indicates the area is below the x-axis. Taking the absolute value gives the area: 4/3 square units.

This example demonstrates how to apply the formula to find the area between two curves.