Calculus > How to find total distance calculus

Most distance problems in calculus give you the **velocity equation**, which is the derivative of the **position formula.** The velocity formula is normally presented as a quadratic equation. Find total distance by integrating the velocity formula over the given interval. If the graph dips below the x-axis, you’ll need to integrate two or more parts of the graph and add the absolute values.

## How to Find Total Distance Calculus: Steps

Watch the video or read the steps below:

**Sample problem**: Find the total distance traveled for a time interval of 0 to 5 for the function 2t^{2} -4t +3t -6.

Step 1: **Identify the velocity function** from the question and the given intercepts of time. The velocity function is 2t^{2}-4t + 3t – 6 and the time interval is 0≥ t ≤5

Step 2: **Graph your velocity function** and note where areas of the graph are above or below the x-axis. This where the velocity as a vector has changed direction.

Step 3:

**Integrate the velocity function.**You’ll need to integrate each shaded area on the graph separately — the area

*above*the x-axis and the area

*below*the x-axis. For this particular function, integrate from 0 to 2 and 2 to 5. You

*could*find the definite integral by hand, but the fastest way is to integrate the function using the TI-89. You can find the integration function on the F5: Math menu (TI-89 Titanium), which appears after you have graphed the function.

If you don’t have a TI-89, you can use this widget from Wolfram:

Step 4:

**Add the**absolute value of the areas you calculated in Step 4 to find the total distance. For this particular function, 49.5 + |-8.66667| = 58.1667.

*That’s how to find total distance in calculus!*

Tip: In the case the intervals of the function are completely above x you just have to find the definite integral for one area. So you won’t need to find the absolute value of the function.

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