Area Under the Curve Calculator: Find Area, Formula & Steps

Latest Updates

Book Free Demo :

One-to-One Learning - Any Topic, Any Time

How to Calculate Area Under a Curve with Steps and Examples

Area Under the Curve Calculator

What is Area Under the Curve Calculator?

The Area Under the Curve Calculator helps you find the region between the graph of a function (y = f(x)) and the x-axis over a specific interval. It uses calculus concepts—specifically definite integration—to compute the precise area. This is useful in mathematics, statistics, physics, engineering, and data analysis where it represents quantities like accumulated totals, probabilities, or the total effect.

Formula or Logic Behind Area Under the Curve Calculation

The primary formula used is the definite integral. For a function y = f(x) between limits a and b:

Area = ∫_a^b f(x) dx
In simple terms, integration adds up infinitely small slices under the curve to get the total area. For most functions, the calculator uses Simpson’s Rule—a numerical method that gives a highly accurate estimate even for complicated graphs. For discrete sets of points (data), a summing approach (like Riemann sums) may be used.

Area Under the Curve: Examples Table

Function y = f(x)	Interval [a, b]	Area Under Curve
x	0 to 1	0.5
x^2	0 to 1	0.333
sin(x)	0 to π	2
e^x	0 to 1	1.718

Steps to Use the Area Under the Curve Calculator

Enter your function in terms of x, e.g., x^2, sin(x), 3*x+1
Provide the lower and upper interval limits (e.g., 0 and 2)
Click on the 'Calculate Area' button
View the computed area and stepwise working instantly

Why Use Vedantu’s Area Under the Curve Calculator?

Vedantu’s Area Under the Curve Calculator provides instant, accurate, and stepwise area computations for any mathematical function over an interval. It’s mobile-friendly, simple to use, requires no math software, and is trusted by millions of students for JEE, NEET, CBSE, ICSE, and competitive exam practice. Quickly validate answers and boost confidence in calculus topics!

Real-life Applications of Area Under the Curve

In statistics, the area under a probability curve represents the probability of certain events (e.g., normal distribution, ROC-AUC in machine learning)
In physics, it helps calculate total distance (from velocity-time graph) or work done (from force-distance graph)
Pharmacokinetics uses AUC to measure total drug exposure over time
Finance uses area for revenue/cost analysis over time
In engineering, find material properties using area under stress-strain graphs

The calculator is vital for students, researchers, and professionals looking to solve real-world problems or exam questions easily.

Looking for more math tools? Explore: Definite Integrals, Area of a Curve, Riemann Integral, Taylor Series, Integral Calculus for deeper learning and practice.

FAQs on Area Under the Curve Calculator: Find Area, Formula & Steps

1. What is the area under the curve?

The area under the curve (AUC) represents the total area between a function's graph and the x-axis over a specified interval. It's a fundamental concept in calculus and statistics, used to calculate quantities like total distance traveled or probability.

2. How do I calculate the area under the curve?

The area under a curve is calculated using definite integration. For a function f(x) from x = a to x = b, the AUC is given by the definite integral: ∫_a^b f(x) dx. This integral represents the limit of a Riemann sum, approximating the area with many small rectangles.

3. What is the formula for the area under the curve?

The basic formula for the area under the curve (AUC) of a function f(x) between limits a and b is: ∫_a^b f(x) dx. This means finding the definite integral of the function over the specified interval.

4. What does AUC mean in statistics?

In statistics, AUC often refers to the Area Under the Receiver Operating Characteristic (ROC) curve. This represents the model's ability to distinguish between two classes. A higher AUC indicates better classification performance.

5. How do you find the area under a curve using integration?

To find the area under a curve using integration, first find the indefinite integral of the function. Then, evaluate the indefinite integral at the upper and lower limits of the interval. Finally, subtract the value at the lower limit from the value at the upper limit. This difference gives the area under the curve.

6. What are some applications of the area under the curve?

The area under the curve has numerous applications across various fields. Some examples include calculating total distance traveled from a velocity-time graph, determining probability in statistics, analyzing drug exposure in pharmacokinetics, and assessing model performance in machine learning.

7. How to calculate the area under the normal curve?

Calculating the area under the normal curve often involves using a z-table or statistical software. You standardize the data by converting to z-scores and then look up the corresponding area in a z-table or use a calculator.

8. What is the difference between AUC and ROC curve?

While both involve the term AUC, they are distinct concepts. The AUC (Area Under the Curve) is a general mathematical concept referring to the area under any curve. The ROC curve is a specific graph used in binary classification problems. Its AUC measures the classifier's overall performance.

9. How is the area under the curve used in pharmacokinetics?

In pharmacokinetics, AUC (Area Under the Curve) represents the total drug exposure over time. It's a valuable measure for comparing the bioavailability of different drug formulations or routes of administration. A higher AUC generally indicates greater drug exposure.

10. Can I use Excel to calculate the area under the curve?

Yes, Excel offers functions to approximate the area under a curve. You can use numerical integration methods, such as the trapezoidal rule or Simpson's rule, to estimate the area using the data points of your curve. Alternatively, use numerical integration add-ins.

11. What is the Riemann sum and its relation to area under the curve?

The Riemann sum is a method for approximating the area under a curve by dividing the area into a series of rectangles. The sum of the areas of these rectangles approximates the total area. As the number of rectangles increases, the approximation improves, ultimately converging to the definite integral (and thus the exact area) of the curve.