How does integration find average

calculus

I am slightly new to calculus. I haven't been able to get the concept that how plotting the area under the line gives us average value, not the instantaneous value. Then, how do you find the instantaneous value? I have seen few answers but they are not very beginner friendly.

Best Answer

Here's an intuitive way to look at it:

  1. Suppose the function $f(t)$ represents something physical, like the height of a water level over time. If the graph of $f(t)$ is piecewise constant (made of horizontal lines) like this: piecewise constant function then you can compute its average value over any time period using algebra (no calculus needed). It's a weighted sum: For each height level, you measure the height and multiply it by whatever percent of the time the function is at that height (in other words, the length of time the function is at that height, divided by the total time). Then you add all those weighted values together to get the average height overall.

    For example, if the function is at height 6m for 3 out of 10 total seconds, and at height 21m for 7 out of 10 total seconds, then the average height is 6*0.3 + 21 * 0.7 = 16.5m.

    Your answer has the right physical units: height. And it has the right behavior in extreme cases, like if the function is at height 6 for 99% of the time, the average value will be close to 6.

  2. Another way of writing this weighted sum is like this: for each height, you multiply it by the length of time the function is at that height. You add up all of those values, then divide by the total length of time. This gives your average height just like the earlier formula. To reuse the earlier example, we could also have computed the average as (6*3 + 21*7)/10.

  3. This second formula is convenient because it corresponds to something in the picture. It says that to compute the average height of this function, just compute the area of each of the rectangles. Add up those areas, then divide by the total width of the time interval (!).

  4. We've arrived at a formula: To compute the average height of a function over a certain interval of time, compute the area under that function during that interval, then divide by the length of that interval.

  5. You can think of integration, in part, as a tool for calculating the area under curves that are not made of straight lines. If a curve has the formula $f(t)$, then its integral $\int_{a}^b f(t)\,dt$ tells you the area under that curve.

    Intuitively, the formula for average height should still be the same: compute the area under the curve, then divide by the length of the interval. Turns out, that's exactly right: if $f(t)$ is a function, then its average height over the interval between $t=a$ and $t=b$ is its area divided by the length of the interval, or in symbols:

    $$f_{ave} = \frac{1}{b-a} \int_{a}^b f(t)\,dt$$



(Nitpick: technically the areas are *signed*, so areas under the curve count as negative, but I didn't want to clutter the discussion with that detail.)
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