Find f · dr c for the given f and c. f = x2 i + y2 j and c is the top half of a circle of radius 3 starting at the point (3, 0) traversed counterclockwise. (a) parameterize the path c in the usual way for circles. r (t) = (b) set up the integral in terms of t only. f · dr c = dt (c) evaluate the integral in part (b).

An automated water dispenser fills packets with one liter of water on average, with a standard deviation of 5 milliliter. the manual says that after a year of operation the dispenser should be tested to see if it needs recalibration. a year later a number of filled packets are set aside and measured separately. it is found that the average packet now contains about 0.995 liters. does the dispenser need calibration? explain your answer.

One of the same side angles of two parallel lines is 20° smaller than the other one. find the measures of these two angles.

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.1. (round your answers to four decimal places.) (a) if the distribution is normal, what is the probability that the sample mean hardness for a random sample of 8 pins is at least 51?