Multivariable Calculus – How to Calculate Area of Sphere Inside a Cylinder

Question

The area of ​​the portion of the sphere $ x^{2} + y^{2} +z^{2} = 1$ located inside of the cylinder $x = x^{2} + y^{2}$, and above the plane $z = 0$.

I'm stuck, so any tip will be helpful

Thanks in advance!

Answer 1

Hint:

$$z^2 = 1-x^2-y^2$$

$$S = \int\int_{D} \sqrt{1+f_{x}^2 +f_{y}^2}dA$$

$$ f_x = -\frac{x}{z}, f_y = -\frac{y}{z}, dA = rdrd\theta$$

$$S = \int_{0}^{\pi}\int_{0}^{cos(\theta)} \frac{1}{\sqrt{1-r^2}}rdrd\theta$$

Evaluate this and that will be your answer.