Area under the curve calculator

The area under a curve y = f(x) between x = a and x = b is the definite integral of f from a to b: find an antiderivative F, then work out F(b) − F(a). The area under y = x² + 1 from 0 to 2 is [x³/3 + x] from 0 to 2 = 14/3, about 4.667 square units. This area under curve calculator is the Integral Calculator set to Definite: it shows the working, shades the area on the graph, and gives the total area too when part of the curve dips below the x-axis.

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Integral Calculator

Integrate step by step and shade the area under the curve

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How to find the area under a curve in the calculator

1. Open the Integral Calculator. It opens on Definite, which is the mode for areas. (Indefinite gives the antiderivative only.)
2. Type the curve in f(x), such as x^2+1. Use \sin, e^x, \ln and fractions as you would write them.
3. Type the limits in From and To: numbers, values like \pi/2, or infinity.
4. Read the definite integral, exact where it can be, with its decimal value, the antiderivative and the working.
5. Check Area, all counted positive. The graph shades the area above the axis in one colour and below it in another. Drag the limit markers on the graph to move them.

The area under a curve formula

This works directly when the curve stays above the x-axis between a and b. Where the curve is below the axis, the integral counts that area as negative. So the definite integral is the net area: area above minus area below. For the total area, split the integral where the curve crosses the axis and add the sizes of the pieces.

Worked examples you can check in the calculator

Example 1: a curve above the axis

Find the area under y = x² + 1 from x = 0 to x = 2. Type x^2+1, from 0, to 2.

The calculator gives 14/3, a numeric value of 4.6667, and the same total area, since none of it is below the axis.

Example 2: a curve below the axis

y = x² − 5x is below the x-axis between its roots, 0 and 5. Type x^2-5x from 0 to 5:

The integral is −125/6 ≈ −20.833, negative because the region is below the axis. An area can't be negative, so the area is 125/6 ≈ 20.833 square units, which is what Area, all counted positive shows.

Example 3: net area and total area

Integrate sin x from 0 to 2π. The calculator gives exactly 0: the arch above the axis from 0 to π cancels the arch below it from π to 2π. Each arch has area [−cos x] from 0 to π = 1 − (−1) = 2, so the total area is 4, which is what Area, all counted positive shows: it splits the integral where the curve crosses the axis and adds the sizes of the pieces.

Here x is the angle in radians and y = sin x. By hand, find where the curve crosses the axis, integrate each piece separately and add their sizes: |2| + |−2| = 4.

Example 4: the area between two curves

Find the area enclosed by y = 4 − x² and y = x + 2. They meet where 4 − x² = x + 2, so x² + x − 2 = 0, (x + 2)(x − 1) = 0, and x = −2 or x = 1. Between them, the parabola is on top, so integrate top minus bottom: (4 − x²) − (x + 2) = 2 − x − x². Type 2-x-x^2 from −2 to 1:

The area is 4.5 square units. You can also type the difference as it stands, (4-x^2)-(x+2), and the calculator gives the same 9/2, with the brackets kept in its heading.

Here y is the gap between the two curves at each x. The area under this one curve, between its roots, is the area between the original two.

Example 5: distance from a velocity graph

A car slows down with velocity v = 12 − 3t m/s, stopping at t = 4 s. Its distance travelled equals the area under its velocity-time graph, so type 12-3x from 0 to 4 (the calculator uses x for the variable): ∫(12 − 3t) dt = [12t − 1.5t²] from 0 to 4 = 48 − 24 = 24 m.

Example 6: when there is no formula

e^x² has no antiderivative in standard functions. Type e^{x^2} from 0 to 1: the calculator says no antiderivative was found and gives the area by numeric integration, ≈ 1.4627.

Approximating the area: the trapezoidal rule

The calculator doesn't do Riemann sums or the trapezoidal rule (the trapezium rule in the UK), so here they are by hand. Split [a, b] into n strips of width h = (b − a) ÷ n, read off the heights y₀, y₁, … yₙ, and treat each strip as a trapezoid (trapezium):

For e^x² from 0 to 1 with 4 strips, h = 0.25 and the heights are 1, 1.0645, 1.2840, 1.7551 and 2.7183. The rule gives 0.125 × (1 + 2 × 4.1036 + 2.7183) = 1.4907. Compare it with the calculator's 1.4627: the trapezoids sit above a curve that bends upwards, so the rule overestimates. More strips bring it closer.

Common mistakes

• Taking a negative integral as the area. Areas are positive. Give the size.
• Integrating straight across a root. If the curve crosses the axis between the limits, the pieces cancel. Split at the root for total area.
• Subtracting the wrong way. It is F(upper) − F(lower).
• Bottom minus top for the area between curves. Integrate the top curve minus the bottom one, or take the size of the answer.
• Degrees in trig integrals. Limits like π/2 are in radians. Type \pi, not 90.

Read next

Maths · CalculusWhat is an integral? Meaning, area and worked examplesWhat an integral is: area under a curve, adding up tiny pieces, and the reverse of differentiation. Definite and indefinite integrals with worked examples.