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 2t2 -4t +3t -6.
Step 1: Identify the velocity function from the question and the given intercepts of time. The velocity function is 2t2-4t + 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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