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The video begins with finite samples, turns their average into an integral and finds the average of sin x on [0,π]. It then relates average tangent slope to an antiderivative’s endpoint change.
- 1A finite average becomes an integralEven spacing connects the sample count to Δx and interval width.
- 2Average sine heightThe area is 2, so the average on [0,π] is 2/π.
- 3Average tangent slopeThe average derivative equals the antiderivative’s endpoint change divided by width.
- 4Signed areaValues below the axis contribute negatively to the integral.
ポイント
- A continuous average can be approached with finite, evenly spaced samples.
- Average value is the integral divided by interval length.
- The average of sin x on [0,π] is 2/π.
- An antiderivative turns the average derivative into an endpoint secant slope.
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A continuous average
3Blue1Brown uses the average value of a continuous function to give another perspective on the relationship between integrals and derivatives. The example is the height of sin x on the interval from 0 to π.
3Blue1Brown uses the average value of a continuous function to give another perspective on the relationship between integrals and derivatives. The example is the height of sin x on the interval from 0 to π.
Sine waves can model cyclic behavior, so an average over part of a cycle can be useful. The video simplifies the application to the pure sine function before adding real-world constants.
A finite average adds values and divides by their count. There are infinitely many inputs in a continuous interval, so the expression cannot simply be an ordinary sum divided by infinity.
From samples to area
Instead, sample a finite number of evenly spaced points and average their heights. As the spacing shrinks, these finite averages approach the continuous average for this well-behaved function.
Instead, sample a finite number of evenly spaced points and average their heights. As the spacing shrinks, these finite averages approach the continuous average for this well-behaved function.
An integral also starts from a finite approximation, but adds height times width: sin x Δx. The integral is the limit of these sums as the partition becomes finer.
With sample spacing Δx over a width of π, the number of samples is approximately π/Δx. Substituting this into the finite average turns its numerator into a sum of heights times Δx.
Taking the limit gives the integral of sin x from 0 to π divided by π. Geometrically, it is the area under the curve divided by the interval’s width: an average height.
Evaluating the sine integral
To evaluate the integral, use an antiderivative: the derivative of −cos x is sin x. Its value changes from −1 at zero to 1 at π.
To evaluate the integral, use an antiderivative: the derivative of −cos x is sin x. Its value changes from −1 at zero to 1 at π.
The change in the antiderivative is 2, so the integral is 2 and the average height is 2/π, approximately 0.64. The average is lower than the peak height of 1.
Average slope
The same calculation can be read as a slope: the change in −cos x divided by the change in x is the slope of the secant line between the endpoints.
The same calculation can be read as a slope: the change in −cos x divided by the change in x is the slope of the secant line between the endpoints.
At each point, sin x gives the tangent slope of −cos x. Averaging those slopes over the interval produces the overall change divided by the interval length.
The general rule
For a general integrable function f on [a,b], its average value is (1/(b−a)) times the integral from a to b. Areas below the axis contribute negatively, so the integral is a signed area.
For a general integrable function f on [a,b], its average value is (1/(b−a)) times the integral from a to b. Areas below the axis contribute negatively, so the integral is a signed area.
If F is an antiderivative of f under the usual conditions, the fundamental theorem of calculus gives the integral as F(b) − F(a). Dividing by b − a gives the secant slope of F.
Adding a constant to an antiderivative moves its graph vertically but leaves the endpoint difference unchanged. The average slope therefore does not depend on which vertical shift is chosen.
When to look for an integral
The final lesson is a useful signal for integration: when a finite idea involves summing many values and must be extended to a continuous range, approximate it with finite sums and examine the limit.
The final lesson is a useful signal for integration: when a finite idea involves summing many values and must be extended to a continuous range, approximate it with finite sums and examine the limit.
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