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Learn how to calculate the area under the curve using different methods, such as rectangles, integration, and antiderivative. Find formulas for various curves, such as circle, parabola, and ellipse, with respect to x-axis or y-axis.
- Integration
Integration is the process of finding the area of the region...
- Ellipse
Area of Ellipse Formula. The area of an ellipse is defined...
- Integration
Learn how to calculate the area under the curve of a function using definite integrals and antiderivatives. See examples of cases where the area is above, below, or partly on the x-axis.
- Overview
- Summary
- Detailed Example
- Sample Problems
The formula to determine the area under a curve, plus lots of helpful examples.
While finding the area under a curve in calculus isn’t as straightforward as finding the area of various geometric shapes, it’s not as difficult as you might fear! We’re here to help you understand the question, choose the right formula, plug in the data, and calculate the result. Check out the detailed example for a step-by-step walk-through, then move onto the sample problems to help build your skills.
Plug your given function into the formula
A = ∫a,b f (x) dx
Use the power rule [
∫a,b f (x) dx = (n^x+1)/ (n+1) |a,b
Find the area under a curve by doing a definite integral with
A = ∫a,b f (x) dx
This is the standard formula you’ll use to determine the area under a curve. It tells you that you’re finding the differential (
) of the integral (
) of a function (
indicates that this is a definite (not indefinite) integral with known limits (for example, if you’re asked to find the area between
Sample Problem—Find the area under
Expect the question to be phrased something along these lines. To help you visualize the curve and the area you need to determine, consider
If you do graph it, you’ll see that
looks like a big smiley face with its vertex (the point where the curve changes direction) at
For this sample problem, you need to calculate the area (in square units) between the vertex (
) and an imaginary straight line drawn from
A = ∫a,b f (x) dx
A = ∫1,3 x^3 dx
A = (3^4)/4 - (1^4)/4
A = 81/4 - 1/4 = 80/4
A = 20 square units
A = ∫a,b f (x) dx
Learn how to use integrals to find the area under a curve or between two curves. See video lessons, solutions, formulas, diagrams and interactive tools for calculating and approximating areas.
Learn how to find the area under a curve using integration, with examples and diagrams. See how to deal with different cases of curves above, below or crossing the x-axis.
- In the diagram above, a "typical rectangle" is shown with width Δx\displaystyle\Delta{x}Δx and height y\displaystyle{y}y. Its area is yΔx\displayst...
- We consider the case where the curve is below the x\displaystyle{x}x-axis for the range of x\displaystyle{x}x values being considered.In this case,...
- In this case, we have to sum the individualparts, taking the absolute value for the section where the curve is below the x\displaystyle{x}x-axis (f...
- In some cases, it is easier to find the area if we take vertical sums. Sometimes the only possible way is to sum vertically.In this case, we find t...
Finding the Area Under a Curve. The area under a curve can be approximated with rectangles equally spaced under a curve as shown below. For consistency, you can choose whether the boxes should hit the curve on the left hand corner, the right hand corner, the maximum value, or the minimum value.
Calculate the area under any curve using this free online tool. Enter the function, the interval and the number of subintervals, and get the exact answer and the step-by-step solution.
