Use stokes' theorem to evaluate c f · dr where c is oriented counterclockwise as viewed from above. f(x, y, z) = xyi + 5zj + 7yk, c is the curve of intersection of the plane x + z = 1 and the cylinder x2 + y2 = 81.

Step-by-step explanation:

The given curve is .

Let us parameterize C to obtain a vector equation for C.

We have ,   .

This is the equation of a circle( a cylinder in space) with radius, r=9 units.

The parametric equation of a circle is given by:

and

This implies that:

and

From, , we obtain, ,

A vector equation for C is;

for

Also, we have;

Since C is oriented counterclockwise as viewed from above, we can apply the Stokes' Theorem.

Thus, by Stokes' Theorem;

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37/3

step-by-step explanation:

step 1: simplify both sides of the equation.

7x+8(x+

1

4

)=3(6x−9)−8

7x+(8)(x)+(8)(

1

4

)=(3)(6x)+(3)(−9)+−8(distribute)

7x+8x+2=18x+−27+−8

(7x+8x)+(2)=(18x)+(−27+−8)(combine like terms)

15x+2=18x+−35

15x+2=18x−35

tep 2: subtract 18x from both sides.

15x+2−18x=18x−35−18x

−3x+2=−35

step 3: subtract 2 from both sides.

−3x+2−2=−35−2

−3x=−37

step 4: divide both sides by -3.

−3x

−3

=

−37

−3

x=

37

3