The February 1, 2006 PoW
Consider the Cartesean plane (the xy-plane).
(a) If a line is drawn from
(t, 0) to (1, t) as t runs from 0 to 1, find the area of the figure formed.
{So when t = 0.4, a segment is drawn from (0.4, 0) to (1, 0.4).}
(b) If instead of each line segment being drawn from (t, 0) to (1, t), the segment is drawn vertically, find the area of the figure formed (as t runs from 0 to 1).
{So when t = 0.4, a segment whose length is the distance from (0.4, 0) to (1, 0.4) = sqrt[0.62 + 0.42] = sqrt(0.52) is drawn vertically.}
(c) Finally, consider a segment of fixed length 1 that is initially lying on the x-axis from (0, 0) to (1, 0). If the segment has its right
endpoint at (1, 0) lifted straight up to (1, 1), while the left endpoint is dragged horizontally
along the x-axis until the entire segment is vertical, going from (1, 0) to (1, 1),
find the area of the region that is swept out.
{So when t = 0.3, a segment is drawn from (0.3, 0) to (1, sqrt(0.51)) because the distance from (0.3, 0) to (1, sqrt(0.51)) = sqrt[0.72 + 0.51] = 1.}
