Integral Graph and Area Visualizer

Math Calculus • Integrals

Written by STEM Calculators Team Published June 15, 2026 Updated June 15, 2026

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Theory

What does a definite integral represent?

A definite integral measures net signed area under a graph.

\[ \begin{aligned} \int_a^b f(x)\,dx \end{aligned} \]

Area above the \(x\)-axis counts as positive. Area below the \(x\)-axis counts as negative.

A definite integral is not always the same as total geometric area.

Net signed area

If a graph stays above the \(x\)-axis on \([a,b]\), then:

\[ \begin{aligned} \int_a^b f(x)\,dx \end{aligned} \]

is the ordinary area under the curve.

If the graph goes below the \(x\)-axis, that part contributes negatively:

\[ \begin{aligned} \text{net signed area} &= \text{area above the axis} - \text{area below the axis} \end{aligned} \]

Total geometric area

Total geometric area counts all area as positive. It is found by integrating the absolute value:

\[ \begin{aligned} \text{total area} &= \int_a^b |f(x)|\,dx \end{aligned} \]

This is different from net signed area when the graph crosses the \(x\)-axis.

For example, if the positive and negative areas are equal, the net signed area is \(0\), but the total geometric area is not \(0\).

Accumulated change

The accumulated change function is:

\[ \begin{aligned} A(x) &= \int_a^x f(t)\,dt \end{aligned} \]

This tells how much signed area has accumulated from the starting point \(a\) to the current point \(x\).

At the upper bound \(b\), the accumulated change is the definite integral:

\[ \begin{aligned} A(b) &= \int_a^b f(t)\,dt \end{aligned} \]

Riemann sums

A Riemann sum approximates area by dividing the interval into rectangles.

Let:

\[ \begin{aligned} \Delta x &= \frac{b-a}{n} \end{aligned} \]

where \(n\) is the number of subintervals.

The interval is divided into points:

\[ \begin{aligned} x_0=a,\quad x_1=a+\Delta x,\quad \ldots,\quad x_n=b \end{aligned} \]

Left Riemann sum

A left Riemann sum uses the left endpoint of each subinterval.

\[ \begin{aligned} L_n &= \Delta x \sum_{i=0}^{n-1} f(x_i) \end{aligned} \]

If the function is increasing, the left sum usually underestimates the area. If the function is decreasing, it usually overestimates the area.

Right Riemann sum

A right Riemann sum uses the right endpoint of each subinterval.

\[ \begin{aligned} R_n &= \Delta x \sum_{i=1}^{n} f(x_i) \end{aligned} \]

If the function is increasing, the right sum usually overestimates the area. If the function is decreasing, it usually underestimates the area.

Midpoint Riemann sum

A midpoint Riemann sum uses the midpoint of each subinterval.

\[ \begin{aligned} M_n &= \Delta x \sum_{i=1}^{n} f\!\left(\frac{x_{i-1}+x_i}{2}\right) \end{aligned} \]

The midpoint rule is often more accurate than left or right sums with the same number of rectangles.

Trapezoidal rule

The trapezoidal rule approximates area with trapezoids instead of rectangles.

\[ \begin{aligned} T_n &= \frac{\Delta x}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1}f(x_i) + f(x_n) \right] \end{aligned} \]

It connects neighboring points on the graph with straight line segments.

Simpson’s rule

Simpson’s rule approximates the graph using parabolic arcs. It usually gives a very accurate result for smooth functions.

Simpson’s rule requires an even number of subintervals.

\[ \begin{aligned} S_n &= \frac{\Delta x}{3} \left[ f(x_0) + 4\sum_{\text{odd }i}f(x_i) + 2\sum_{\text{even }i}f(x_i) + f(x_n) \right] \end{aligned} \]

If \(n\) is odd, a calculator may use the next even value of \(n\).

Worked example: \(\int_0^\pi \sin(x)\,dx\)

Consider:

\[ \begin{aligned} \int_0^\pi \sin(x)\,dx \end{aligned} \]

The graph of \(\sin(x)\) is above the \(x\)-axis on \([0,\pi]\), so the integral is positive.

An antiderivative is:

\[ \begin{aligned} \int \sin(x)\,dx &= -\cos(x)+C \end{aligned} \]

Therefore:

\[ \begin{aligned} \int_0^\pi \sin(x)\,dx &= [-\cos(x)]_0^\pi\\ &= -\cos(\pi)-[-\cos(0)]\\ &= 1-(-1)\\ &= 2 \end{aligned} \]

So the exact net signed area is \(2\).

When net area cancels

Sometimes positive and negative areas cancel.

For example:

\[ \begin{aligned} \int_0^{2\pi}\sin(x)\,dx &= 0 \end{aligned} \]

The positive area from \(0\) to \(\pi\) cancels the negative area from \(\pi\) to \(2\pi\).

But the total geometric area is not \(0\), because there is still area between the graph and the \(x\)-axis.

Common mistakes

Confusing net area with total area: net area can be zero even when the graph encloses visible area.
Ignoring negative regions: regions below the \(x\)-axis subtract from the definite integral.
Using too few rectangles: a small value of \(n\) gives a rough approximation.
Using Simpson’s rule with odd \(n\): Simpson’s rule needs an even number of subintervals.
Forgetting \(\Delta x\): each Riemann sum must multiply function values by the subinterval width.
Thinking the accumulated graph is the original function: \(A(x)\) is an integral of \(f(x)\), not the same graph as \(f(x)\).

Frequently Asked Questions

What does net signed area mean?

Net signed area means that area above the x-axis counts positive and area below the x-axis counts negative.

What is the difference between net area and total area?

Net area subtracts area below the x-axis, while total geometric area counts both above and below as positive.

What does the accumulated change graph show?

It shows A(x) = integral from a to x of f(t) dt, so each point gives the accumulated signed area from the lower bound to that x-value.

What is a left Riemann sum?

A left Riemann sum uses the left endpoint of each subinterval as the rectangle height.

What is a right Riemann sum?

A right Riemann sum uses the right endpoint of each subinterval as the rectangle height.

What is a midpoint Riemann sum?

A midpoint Riemann sum uses the midpoint of each subinterval as the rectangle height.

What does the trapezoidal rule do?

The trapezoidal rule approximates the area under the curve using trapezoids instead of rectangles.

Why does Simpson’s rule sometimes use a different n?

Simpson’s rule requires an even number of subintervals. If n is odd, the calculator uses the next even number.

Can I use pi in the bounds?

Yes. You can use pi, pi/2, 2pi, and similar expressions.

Is the shaded area exact?

The shading is a visual representation. The displayed integral value is calculated numerically using high-resolution Simpson approximation.

Integral Graph and Area Visualizer

Function and interval

Graph and output settings

Quick examples

Integral area graph

Step-by-step area interpretation

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Frequently Asked Questions

Function and interval

Graph and output settings

Quick examples

Integral area graph

Step-by-step area interpretation

Rate this calculator

Frequently Asked Questions

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