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Calculate Area Between Two Curves Online

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How to Find Area Between Two Curves: Explained with Formula & Examples

Area Between Two Curves Calculator – Free Online Tool with Formula, Steps & Examples | Vedantu

Area Between Two Curves Calculator

What is Area Between Two Curves?

The area between two curves calculator finds the region that lies between the graphs of two functions, usually over a specific interval [a, b] along the x-axis. Graphically, if you have two curves, the area between them represents the "space" trapped vertically between the top and bottom curves. In calculus and geometry, this is a key application of definite integrals to quantify the bounded region.

Formula or Logic Behind Area Between Two Curves Calculator

To calculate the area between two curves y = f(x) and y = g(x) over the interval [a, b], use this integral formula:

Area = ∫_a^b [f(x) - g(x)] dx, where f(x) is above g(x).

Steps:
1. Find which function is on top (f(x)) and which is on bottom (g(x)) between x = a and x = b.
2. Subtract the lower function from the upper.
3. Integrate the resulting function from a to b.
4. The answer is the enclosed area (always reported as positive for real-world problems).

For area with respect to y, use x = f(y) and x = g(y), integrating dy between limits.

Sample: Area Between Two Curves – Solved Examples

Function 1, f(x)	Function 2, g(x)	Limits [a, b]	Area
x²	x	0 to 1	1/6 ≈ 0.166667
cos(x)	sin(x)	0 to π/2	1
e^x	0	0 to 1	e - 1 ≈ 1.718282
2x + 1	x	1 to 3	7

Steps to Use the Area Between Two Curves Calculator

Enter the first function f(x) (upper curve) and the second function g(x) (lower curve).
Input the lower and upper limit (a and b) for x.
Click on the 'Calculate Area' button.
The instant solution and formula will appear below.

Why Use Vedantu’s Area Between Two Curves Calculator?

Vedantu’s Area Between Curves Calculator is easy to use, mobile-friendly, and provides fast, accurate solutions. Its simple design helps students learn concepts and get reliable answers instantly. Trusted by lakhs of learners and teachers for school, board, and competitive exam preparation.

Real-life Applications of Area Between Two Curves Calculator

The area between curves appears across academics (board/integral calculus problems), engineering (work under force curves), economics (consumer-producer surplus on supply-demand graphs), probability/statistics (area between probability distributions), geography (land area between boundaries), and science labs. It’s directly tested in CBSE, ICSE, JEE, and other entrance exams. Whether solving homework or analyzing real data, this tool makes calculations fast and error-free.

For more maths help, check out topics like Definite Integral, Algebra, or explore Prime Numbers and Factors for deeper practice.

FAQs on Calculate Area Between Two Curves Online

1. What is the area between two curves?

The area between two curves represents the region bounded by the graphs of two functions, f(x) and g(x), between specified x-values (limits). It's the space enclosed between these curves when plotted on the same coordinate plane. Visually, it's the area 'trapped' between the higher and lower function graphs.

2. What is the formula for finding the area between two curves?

The fundamental formula for calculating the area between curves f(x) and g(x) from x = a to x = b, where f(x) ≥ g(x), is: Area = ∫_a^b [f(x) - g(x)] dx. This represents the definite integral of the difference between the top function and the bottom function, evaluated over the given interval. For functions of y, Area = ∫_c^d [f(y) - g(y)] dy applies.

3. How do I find the area between curves with respect to y?

If the curves are defined as functions of y (x = f(y) and x = g(y)), then the area between them from y = c to y = d, where f(y) ≥ g(y), is calculated using: Area = ∫_c^d [f(y) - g(y)] dy. This involves integrating with respect to y instead of x, and the limits of integration are now y-values.

4. How to find the area enclosed between two curves?

To find the area between two curves, first identify the functions f(x) and g(x) and the interval [a, b] where you want to calculate the area. Determine which function is greater over the interval; it will be the top curve. Then, apply the formula: Area = ∫_a^b [f(x) - g(x)] dx, substituting your functions and limits. Solve the definite integral to find the area. Remember to adjust your method if the functions are defined in terms of y instead of x.

5. What are some real-life applications of finding the area between curves?

The concept of area between curves has diverse applications. In physics, it helps calculate work done by a variable force. In economics, it's used to determine consumer and producer surplus in supply and demand analysis. In statistics, the area can represent probabilities in probability density functions. Other applications include finding the area of irregularly shaped land or regions.

6. What if the curves intersect within the interval [a, b]?

If the curves intersect within the interval [a, b], you need to find the intersection points. Then, you split the integral into multiple integrals, with each integral covering a sub-interval where one curve consistently lies above the other. You'll calculate the area for each sub-interval separately and sum the results to obtain the total area.

7. How do I use the area between two curves calculator?

To use the area between two curves calculator, input the equations of the two functions, f(x) and g(x). Next, enter the lower limit (a) and upper limit (b) of the interval. Click the 'Calculate' button. The calculator will compute the definite integral and display the calculated area between the curves.

8. What happens if the curves do not intersect?

If the curves do not intersect within the specified interval [a, b], you simply apply the standard formula Area = ∫_a^b [f(x) - g(x)] dx, where f(x) is the upper function and g(x) is the lower function across the entire interval. No splitting of the integral is needed in this case.

9. Can this calculator handle trigonometric functions?

Yes, the calculator can typically handle various functions, including trigonometric functions (sin(x), cos(x), tan(x), etc.). Make sure you input these functions correctly using the appropriate syntax, as defined in the calculator's input guidelines.

10. What if my functions are more complex?

For more complex functions, ensure you enter the functions accurately, using the correct syntax for the calculator. You may need to simplify your functions before inputting them, or use a separate tool to check for errors before using the area between curves calculator. The accuracy of the result depends on the accurate input of the functions and the limits.

11. Is there a way to check my answer after using the calculator?

To check your answer, you can manually calculate the definite integral using integration techniques and compare the result to the calculator's output. This will verify that the calculator’s result is correct. You can also cross-check your results by using another area calculator or reviewing solved examples.