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Area Under Curve Calculator

Calculate the area under any curve between two x-values with a shaded visualization showing positive and negative regions.

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Basic Polynomial

Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.

Key values: x^2 · [0, 1] · Simpson's rule

Trigonometric Integral

Integrate sin(x) from 0 to pi. Exact answer: 2.

Key values: sin(x) · [0, pi] · Exact: 2

Area Under a Bell Curve

Approximate the Gaussian integral e^(-x^2) from -3 to 3.

Key values: e^(-x^2) · [-3, 3] · Gaussian

Documentation

Area Under a Curve

The area between a non-negative function and the x-axis from $x = a$ to $x = b$ is:

A = \int_a^b f(x) \, dx

If the function dips below the x-axis, use $|f(x)|$ for the geometric (unsigned) area.

Area Between Two Curves

The area between $f(x)$ (upper) and $g(x)$ (lower) from $a$ to $b$ :

A = \int_a^b [f(x) - g(x)] \, dx

Key step: Find intersection points by solving $f(x) = g(x)$ — these are the limits of integration. If the curves cross within the interval, split into separate integrals.

Worked Example

Find the area between $y = x^2$ and $y = x$ :

Intersection: $x^2 = x \implies x(x-1) = 0 \implies x = 0, 1$
On $[0, 1]$ , $x \geq x^2$ (line is above parabola)
$A = \int_0^1 (x - x^2) \, dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

Horizontal Slicing

Sometimes integrating with respect to $y$ is easier. Express curves as $x = f(y)$ and $x = g(y)$ :

A = \int_c^d [f(y) - g(y)] \, dy

This is especially useful for regions bounded by curves that aren't functions of $x$ (like a sideways parabola $x = y^2$ ).

Frequently Asked Questions

How do you calculate the area under a curve?

The area under a non-negative curve $f(x)$ from $x = a$ to $x = b$ is computed by the definite integral $\int_a^b f(x)\\,dx$ . If the function dips below the x-axis, use $|f(x)|$ for the geometric (unsigned) area.

What is the difference between area under a curve and a definite integral?

A definite integral gives the signed area, where regions below the x-axis count as negative. The geometric area under a curve treats all regions as positive by integrating the absolute value $|f(x)|$ .

How do you find the area between two curves?

Find the intersection points by solving $f(x) = g(x)$ to get the limits. Then integrate the difference: $\int_a^b [f(x) - g(x)]\\,dx$ , where $f(x)$ is the upper curve and $g(x)$ is the lower curve.

What if the curves cross within the integration interval?

If the curves cross, split the integral at each intersection point. On each sub-interval, determine which function is on top and integrate the appropriate difference. Then sum the absolute values of each piece.

When should I use horizontal slicing instead of vertical slicing?

Use horizontal slicing (integrating with respect to $y$ ) when the region is bounded by curves that are more naturally expressed as $x = f(y)$ , such as a sideways parabola $x = y^2$ . This can simplify the integral significantly.

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