Area Under the Curve Calculator

Q: How do you find the area under a curve?

Integrate: the area between y = f(x) and the x-axis from a to b is \int_a^b f(x)\,dx, as long as f \ge 0 there. If the curve dips below the axis, split at the zeros and add the absolute values.

Q: How do I find the area between two curves?

Enter both curves. The area is \int_a^b |f(x) - g(x)|\,dx. If you leave the bounds blank-ish, use the intersection points shown as your bounds.

Shade it, then measure it. Net (signed) area and total area side by side, with the region split at every zero, and optional second curve for area between curves.

Top curve f(x)

Reads as

x^{2} - 2 x

Try:

Bottom curve g(x) (0 for the x-axis)

Reads as

0

Try:

From aTo b

Total area

2.666667

Net (signed) area

Split at the zeros

\int_{0}^{2} \left(x^{2} - 2 x\right) dx = -\frac{4}{3}

\int_{2}^{3} \left(x^{2} - 2 x\right) dx = \frac{4}{3}

\text{Total} = \left|-\frac{4}{3}\right| + \left|\frac{4}{3}\right| = \frac{8}{3}

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Net area vs. total area

A definite integral adds area above the axis and subtracts area below it. That is the right answer for questions like displacement or net change. When a question asks for “the area of the region”, it means total area: find where the curve crosses the axis, integrate each piece, and add the absolute values. The calculator does that split for you and shades positive pieces amber and negative pieces red.

Area between two curves

Enter a second curve and the region between them is measured: $A = \int_a^b |f(x) - g(x)|\,dx$ . The typical exam problem, “the region enclosed by $y = x$ and $y = x^2$ ”, uses the intersection points 0 and 1 as bounds and gives $\tfrac16$ .

Each exact value comes from an antiderivative; see the steps on the integral calculator, or estimate with rectangles on the Riemann sum calculator.

Questions students ask

How do you find the area under a curve?

Integrate: the area between $y = f(x)$ and the x-axis from $a$ to $b$ is $\int_a^b f(x)\,dx$ , as long as $f \ge 0$ there. If the curve dips below the axis, split at the zeros and add the absolute values.

What is the difference between net (signed) area and total area?

Net area is the plain definite integral, where area below the axis counts negative. Total area adds $\int_a^b |f(x)|\,dx$ , so every piece counts positive. Distance travelled is total area under a velocity graph; displacement is net area.

How do I find the area between two curves?

Enter both curves. The area is $\int_a^b |f(x) - g(x)|\,dx$ . If you leave the bounds blank-ish, use the intersection points shown as your bounds.

Why is my area negative?

Because more of the region lies below the x-axis than above it. The total area shown next to it is always positive.