These sums are examples of Riemann sums (more specifically, left-hand Riemann sums), which are probably familiar to you as systematic ways to approximate areas under curves. (See, for example, the module on Accumulation.)
Let denote the circular arc joining the points and , traced out counter-clockwise. [Note that this is the opposite
direction to that in Part 4.] One way to describe this arc is via a pair of
What is the sum of these integrals? How is this answer related to the work you calculated in Part 4?
Notice that the function specifies the -coordinate in the parametrization of the curve , and the function specifies the -coordinate in the parametrization of the curve .
In the next two steps, you will
use the force defined by the equation
and the parabolic path along y = 1 - x2 from the point to , as in the following figure.
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