How might Venus’ atmosphere have evolved to its present state through a runaway greenhouse effect?

Assume that no segment of the curve

$$x = x ( t ) , \quad y = y ( t ) , \quad ( a \leq t \leq b )$$

is traced more than once as t increases from a to b. Divide the interval [a, b] into n subintervals by inserting points

$$t _ { 1 } , t _ { 2 } , \ldots , t _ { n - 1 }$$

between

$$a = t _ { 0 }$$

and

$$b = t _ { n }$$

. Let L denote the arc length of the curve. Give an informal argument for the approximation

$$L \approx \sum _ { k = 1 } ^ { n } \sqrt { \left[ x \left( t _ { k } \right) - x \left( t _ { k - 1 } \right) \right] ^ { 2 } + \left[ y \left( t _ { k } \right) - y \left( t _ { k - 1 } \right) \right] ^ { 2 } }$$

If dx/dt and dy/dt are continuous functions for

$$a \leq t \leq b$$

, then it can be shown that as max

$$\Delta t _ { k } \rightarrow 0$$

, this sum converges to

$$L = \int _ { a } ^ { b } \sqrt { \left( \frac { d x } { d t } \right) ^ { 2 } + \left( \frac { d y } { d t } \right) ^ { 2 } } d t$$

Give a brief description of the asteroid belt.