Integral Grapher

About the Integral Grapher

Plot a function together with its cumulative numerical integral from the left edge.

How to use this calculator

1. Enter a function such as \(f(x)=x^2\) or \(f(x)=\sin x\).
2. Choose the interval or left edge where the accumulation should start.
3. View the graph of the original function and the cumulative numerical integral on the same page.
4. Use the displayed values to compare the function at each point with its running total from the left.

The formula explained

The tool computes a cumulative integral, which is the running area under \(f(x)\) from the left edge up to each x-value. In symbols, it approximates \(F(x)=\int_a^x f(t)\,dt\), where \(a\) is the left endpoint.

• x = the input value where the graph and accumulated integral are evaluated
• t = the dummy variable used inside the integral
• a = the left edge, or starting x-value, of the accumulation
• f(x) = the original function being graphed
• F(x) = the cumulative integral from the left edge to x

Step by step method

1. Choose the starting point \(a\) and write the function \(f(x)\).
2. Find the area from \(a\) to a chosen x-value, treating parts above the x-axis as positive and parts below as negative.
3. Add areas as x increases to get the running total \(F(x)\).
4. Graph both \(f(x)\) and \(F(x)\) to compare the shape of the function with its accumulation.

Worked example

Problem. Let \(f(x)=2x\) on the interval from \(x=0\) to \(x=3\). Find the cumulative integral at \(x=3\).

1. Set up the accumulation from the left edge, so \(F(3)=\int_0^3 2t\,dt\).
2. Compute the antiderivative, \(\int 2t\,dt=t^2\).
3. Evaluate from 0 to 3, so \(F(3)=3^2-0^2=9\).

Tips and common mistakes

• If the graph goes below the x-axis, the cumulative integral decreases because that area is negative.
• Be careful about the left edge, since the running total depends on where the accumulation starts.

Frequently asked questions