Lets recall the equation of surface area.

Now we need to find all the neccessary equations to be able to evaluate the integral.

We will plug in , into the plane equation in order to get a line that intersects with the z axis.

Now we are going to set , in the previous equation and solve for .

We now have all the bounds for our double integral

First we need to evaluate the vector field evaluated along the curve. 

Now we need to find the derivative of 

Now we can do the product of  and .

Now we can put this into the integral and evaluate it.

The formula for work is given by

.

Writing our path in parametric equation form, we have

.

Hence

Plugging this into our work equation, we get

.

The line integral of a vector field is given by

So, we must evaluate the vector field on the curve:

Then, we take the derivative of the curve with respect to t:

Taking the dot product of these two vectors, we get

This is the integrand of our integral. Integrating, we get

The line integral of the vector field is equal to

The parameterization (using the corresponding elements of the curve) of the vector field is

The derivative of the parametric curve is

Taking the dot product of the two vectors, we get

Integrating this with respect to t on the given interval, we get

To calculate the line integral of the vector field, we must evaluate the vector field on the curve, take the derivative of the curve, and integrate the dot product on the given interval.

The vector field evaluated on the given curve is

The derivative of the curve is given by

The dot product of these is

Integrating this over our given t interval, we get

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function F itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function F itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function F itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function F itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral