I Need to verify these answers (surface area & parametrization)

I know this is a lot of stuff, but I want to be certain that I have all of this information correct. If there are any errors, I feel like they would come from the first problem or from on the last one. Many thanks to anyone willing to help me out with any of this.

**1)** Find the surface area of the solid obtained by revolving for about the y-axis

Surface Area

**2)** Find an equation relating and . Then sketch the curve C whose parametric equations are given and indicate the direction as increases

**a)** , for all t

My graph looks like an elongated graph which goes through (0,0) and has a horizontal tangent at (-1,-1). The direction of t is going upwards.

**b)** for

not since

My graph for this one is the right half of a circle of radius 1 centered on the origin. The direction of t is going clockwise.

**3)** Let C be the curve with parametrization for all t

Find equations of the tangent lines to C at the points corresponding to and . At what point(s) will the graph have a horizontal tangent line? Also find . Use all this information to sketch the curve C.

for , and equation of the tangent line is

for , m is undefined and the equation of the tangent line is

For the graph I have vertical tangents and x intercepts at (-16, 0), (16, 0), a horizontal tangent at (0, -4). The graph crosses itself at the y-intercept (0, 8), is concave down while above the x-axis and concave up when below. It makes a big loop from left to right.

Any help is enormously appreciated (Wink)(Clapping)