How does the limiting process convert the discrete sum of ring areas into the continuous area of a triangle?

Q: How does the limiting process convert the discrete sum of ring areas into the continuous area of a triangle?

By placing strips of height $2\pi r$ along the radius axis, their tops form a linear graph. As the partition width goes to zero, the step-function approximation converges to the area under the line $y=2\pi x$. This region is a right triangle with base $R$ and height $2\pi R$, yielding area $\frac{1}{2} \cdot R \cdot 2\pi R = \pi R^2$.

By placing strips of height

2\pi r

along the radius axis, their tops form a linear graph. As the partition width goes to zero, the step-function approximation converges to the area under the line

y=2\pi x

. This region is a right triangle with base

R

and height

2\pi R

, yielding area

\frac{1}{2} \cdot R \cdot 2\pi R = \pi R^2

Conditions

Function $f(r) = 2\pi r$ is linear
Partition width $\Delta r \to 0$

Reasoning, step by step

Arrange the rectangular strips side-by-side along the horizontal axis representing radius $r$ .
Note that the height of each strip corresponds to the circumference $2\pi r_i$ .
Observe that as $\Delta r$ shrinks, the jagged top edge of the rectangles smooths out.
Identify the limiting shape formed by the tops of the strips as a straight line passing through the origin.
Calculate the area of the resulting geometric figure: a triangle with vertices at $(0,0)$ , $(R,0)$ , and $(R, 2\pi R)$ .
Apply the triangle area formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .

Example

The script explains: 'Place the strips along the radius axis with heights 2πr... As maximum partition width tends to zero, the sums approach the area beneath the linear circumference function... The limiting region is a triangle with base R and height 2πR, giving πR².'

Common misconceptions

Confusing the area of the rectangle bounding box ( $2\pi R^2$ ) with the actual triangular area ( $\pi R^2$ ).
Assuming the limit applies only to the height, not the arrangement of the strips.

Watch the explanation

YouTubeThe essence of calculus

4:07 – 5:52Watch this moment ↗

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