Let vector f = (6 x^2 y + 2 y^3 + 4 e^x) i + (7 e^{y^2} + 54 x) j . consider the line integral of vector f around the circle of radius a, centered at the origin and traversed counterclockwise.
(a) find the line integral for a=1 .
(b) for which value of a is the line integral a maximum?

Denote the circle of radius by . is simple and closed, so by Green's theorem the line integral reduces to a double integral over the interior of (call it ):

is a circle of radius , so we can write the double integral in polar coordinates as

a. For , we have

b. Let denote the integral with unknown parameter ,

By the fundamental theorem of calculus,

has critical points when

If , then line integral is 0, so we ignore that critical point. For the other two, we would find .

solution:

as, mean is defined as sum of all the observations divided by total number of observations.

and , median is mid value of the data whether arranged in ascending order or descending order.

consider, roadsters:

2,3,4,5,6,7,8,9

mean =

also, median= , because number of observations is even, so median is mean of two mid values.

now, coming to bandits

2,3,4,5,6

mean =  

median = 4, as number of observations is odd.

so, roadsters has greater median as well mean than bandits.

→option a is true about the repairs performed on 2 types of cars:

bandits have lower median and mean than roadsters.