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Formula: ∫[a, b] (f(x) - g(x)) dx

Area Between Curves

The area between two curves is a fundamental concept in calculus that measures the space enclosed by two functions over a given interval. It is calculated by integrating the difference between the upper curve and the lower curve within a specified range. This concept is widely used in various fields, including physics, economics, and engineering, to analyze the differences between two varying quantities. By using the area between two curves, we can find solutions for real-world problems such as determining surplus, work done, or resource allocation. Understanding this technique is essential for solving problems involving comparisons of continuous functions.

How to Find the Area Between Two Curves

To find the area between two curves, follow these steps:

Step 1: Identify the Curves

Identify the functions f(x) and g(x) that represent the upper and lower curves, respectively, within the interval of interest.

Step 2: Determine the Limits of Integration

Find the points of intersection between the two curves to determine the lower limit aaa and upper limit bbb of integration.

Step 3: Set Up the Formula

The formula for the area between two curves is:Area=∫a,b[f(x)−g(x)] dx

Where f(x) is the upper curve and g(x) is the lower curve.

Step 4: Integrate the Difference

Use integration to compute the area by subtracting g(x) from f(x) and evaluating the definite integral from a to b.

Step 5: Calculate the Area

After evaluating the definite integral, the result will be the area between the two curves within the defined limits.

Area Between Two Curves Formula

To find the area between two curves, use the formula:

Area=∫a,b[f(x)−g(x)] dx

Where:

  • f(x) is the upper function
  • g(x) is the lower function
  • a and b are the limits of integration (points where the curves intersect).

Area Between Two Curves Examples

Example 1:

Find the area between the curves f(x)=x2 and g(x)=x from x=0 to x=1.

Solution:
The area is calculated as:Area=∫01[x−x2] dx=[x2/2−x3/3]01=1/2−1/3=1/6

Answer: The area between the curves is 1/6 square units.

Example 2:

Find the area between the curves f(x)=sin⁡(x) and g(x)=cos⁡(x) from x=0 to x=π/4.

Solution:
The area is calculated as:Area=∫0π4[cos⁡(x)−sin⁡(x)] dx=[sin⁡(x)+cos⁡(x)]0π4=√2−1

Answer: The area between the curves is approximately 0.4142 square units.

Example 3:

Find the area between the curves f(x)=ex and g(x)=x2 from x=0 to x=1.

Solution:
The area is calculated as:Area=∫01[ex−x2] dx=[ex−x3/3]01=e−4/3

Answer: The area between the curves is approximately 1.7183 square units.

Example 4:

Find the area between the curves f(x)=3x+2 and g(x)=x2 from x=0 to x=2.

Solution:
The area is calculated as:Area=∫02[(3x+2)−x2] dx=∫02[−x2+3x+2] dx=[−x3/3+3x2/2+2x]02=22/3

Answer: The area between the curves is approximately 7.33 square units.

Example 5:

Find the area between the curves f(x)=x3 and g(x)=x2 from x=0 to x=1.

Solution:
The area is calculated as:Area=∫01[x2−x3] dx=[x3/3−x4/4]01=1/3−1/4=1/12

Answer: The area between the curves is 1\12​ square units.

Can the area between two curves be negative?

No, the area between two curves is always positive. If a calculation gives a negative result, it means the lower curve was subtracted from the upper curve incorrectly, or the limits of integration were swapped.

What is the geometric interpretation of the area between two curves?

The geometric interpretation is the total space enclosed between two curves over a given interval. This represents the difference in height between the curves at each point, integrated over the interval.

What if the curves don’t intersect within the interval?

If the curves don’t intersect within the interval, the area is simply calculated using the formula, with f(x)f(x)f(x) as the upper function and g(x)g(x)g(x) as the lower one for the entire interval.

What if the curves intersect?

If the curves intersect, you need to find the points of intersection and break the interval into sections where one curve is always above the other. You then calculate the area for each section separately.

Is the Area Between Curves Always Positive?

Yes, the area between curves is always positive. If a negative result occurs during integration, it means the curves were subtracted incorrectly. The absolute value of the result should be taken to ensure a positive area is calculated.

What Does Area Under the Curve Represent?

The area under the curve represents the total quantity accumulated over an interval, calculated using integration. In various fields, it can indicate values such as distance, revenue, or substance quantity, depending on the application.

What If One Curve Is Always Above the Other?

If one curve is always above the other throughout the interval, simply subtract the lower curve from the upper curve and integrate over the given limits. This makes the calculation straightforward without needing to split the integral.

What Is the Difference Between Area Under a Curve and Area Between Curves?

The area under a curve refers to the space between a curve and the x-axis, while the area between curves is the space enclosed between two distinct functions. The latter involves subtracting one curve from the other.

Why Is Finding the Area Between Curves Important?

Finding the area between curves is important in various fields, such as physics, economics, and engineering, as it helps in calculating differences in quantities, like work done, profit, or resource use, represented by the curves.