Evaluate z c y 2 dx + z 2 dy + (1 − x 2 ) dz where c is the quarter of the circle of radius 1 in the xz–plane with center at the origin in the quadrant x ≥ 0, z ≤ 0, oriented counterclockwise when viewed from the positive y–axis, that is, looking in the −y direction.

Step-by-step explanation:

To find:

where, C is the quarter of the circle of radius 1 in the xz plane.

The center at origin.

where, because it is quarter circle.

The circle oriented counterclockwise when viewed from the positive y–axis.

Therefore,

If  x = cos t, then dx = -sin t dt

If  z = sin t, then dz = cos t dt

If y = 0 , then dy = 0

Substitute the value into integral.

Let

Change the integral limit and variable

Hence, the value of integral is

6, 10

9, -13

12, -21

18, -23

step-by-step explanation:

you just add 6 to the x-coordinate and subtract 10 from the y-coordinate.