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

32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864

27, 54, 81, 108, 135, 162, 189, 216, 243, 270, 297, 324, 351, 378, 405, 432, 459, 486, 513, 540, 567, 594, 621, 648, 675, 702, 729, 756, 783, 810, 837, 864

LCM of 32 and 27 is B. 864

Hope this helps. :)