Projecting a parametric curve to a plane

Question

I have a parametric equation: x = t^3 and y = t + 2t. I would like to do a line integral of this curve up to the plane z = 5.

Basically, I would like to find the area of the "walls" formed when projecting this curve up to the plane z = 5. Let's say we are considering the bounds from t = 0 to t = 1.

Would it simply be the line integral of the parametric curve over the bounds multiplied by 5?

Answer 1

Your curve lies in the $xy$-plane, and so the walls you mention form a right-cylinder.

The area will be the height times the length of the base.

For a parametric curve, the length of the base is given by

$$\int_{t_1}^{t_2} \sqrt{\left(\frac{\mathrm d x}{\mathrm d t}\right)^2+\left(\frac{\mathrm d y}{\mathrm d t}\right)^2}~\mathrm d t$$

Here $t_1$ and $t_2$ are the start and end values of $t$.