Evaluate the double integral ∬r(2x−y)da, where r is the region in the first quadrant enclosed by the circle x^2+y^ 2= 4 and the lines x = 0 and y = x, by changing to polar coordinates.

Step-by-step explanation:

We are to integrate the function

over the area enclosed by the circle

Convert into polar coordinates

x=rcost and y = rsint

Since the lines are x=0 and y=x we find that t varies from 0 to pi/4

Double integral will become now

(since variables are independent here)

Substitute for r and t

4 sin pi/4 -2 cospi/4

=

the length of o'p' is .

step-by-step explanation:

it is given that the quadrilateral m'n'o'p' is the image of quadrilateral mnop after dilation.

the center of dilation is origin and the scale factor is 1.5.

in dilation the image and preimge are similar figures. it means the sides of image and preimage are proportional.

the scale factor is defined as

the scale factor is 1.5.

multiply both sides by op.

note: if the value of op is given, then multiply the value of op with 1.5 to get the length of side o'p'.

therefore length of o'p' is .