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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
The area under the curve can be calculated through three simple steps. First, we need to know the equation of the curve (y = f (x)), the limits across which the area is to be calculated, and the axis enclosing the area. Secondly, we have to find the integration (antiderivative) of the curve.
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This means that when we want to find the area below $f(x)$’s curve and bounded by $x =a$ and $x =b$ as well as the $x$-axis, simply evaluate $f(x)$’s definite integral for the interval, $[a, b]$. How to find the area under a curve? When calculating the area under the curve of $f(x)$, use the steps below as a guide:
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Use the following Definite Integral Calculator to find the Area under a curve. Enter the function, lower bound and upper bound. How to Find Areas Between Curves? Example: Find the area bounded by the curves y = x 2 - 4x and y = 2x. Show Video Lesson
