The accompanying figure shows a spherical cap of height h...

The accompanying figure shows a spherical cap of height $h$ cut from a sphere of radius $r$. Show that the surface area $S$ of the cap is $S=2 \pi r h .$ [Hint: Revolve an appropriate portion of the circle $x^{2}+y^{2}=r^{2}$ about the $y$ -axis.

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Key Concepts

Parametrization of Curves

Parametrization simplifies the process of expressing curves by introducing parameters to describe the coordinates of points on the curve. By converting the problem into one involving these parameters, it becomes easier to set up and solve integrals for quantities like arc length and surface area.

Spherical Cap

A spherical cap is the portion of a sphere that is cut off by a plane. Its geometry is defined by the sphere's radius and the cap’s height, and it appears in various applications such as physics and engineering. Understanding a spherical cap is essential for computing surface areas and volumes of sections of spheres.

Surface Area of Revolution

This concept involves calculating the area of a surface created by rotating a curve about an axis. The method employs integral calculus to sum the contributions of infinitesimally small surface elements, using formulas that incorporate the distance the curve travels during the rotation.

Integral Calculus

Integral calculus is fundamental for determining continuous sums, such as areas under curves and surface areas of objects. In problems involving surfaces of revolution, integrals are set up to calculate the cumulative area of differential surface elements, bridging geometric intuition with analytical techniques.

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