1.5.36. A spherical cap is the set of points on a sphere of radius R whose distance (measured along the sphere) is at most r from a point A on the sphere. (See Figure 1.87.) (a) Find the circumference of the spherical cap in terms of r and R. (b) (Calculus) Verify that the area of the spherical cap is 2n R² - 2 R² cos(r/R). Hint: Use radians. Let x be as shown in Figure 1.87. Recall that the area of the surface obtained by revolving y = f(x) between x = a and x = b about the x-axis is given by fo2n f(x)√1+(f'(x))²dx. Calculate the integral for a spherical cap for general a and b. Then determine a and b in terms of r and R. (c) Archimedes proved that the surface area of the spherical cap is equal to the area of a circle whose radius equals the (straight-line) distance from A to a point on the circumference of the cap. Verify Archimedes' theorem. R 20
1.5.36. A spherical cap is the set of points on a sphere of radius R whose distance (measured along the sphere) is at most r from a point A on the sphere. (See Figure 1.87.) (a) Find the circumference of the spherical cap in terms of r and R. (b) (Calculus) Verify that the area of the spherical cap is 2n R² - 2 R² cos(r/R). Hint: Use radians. Let x be as shown in Figure 1.87. Recall that the area of the surface obtained by revolving y = f(x) between x = a and x = b about the x-axis is given by fo2n f(x)√1+(f'(x))²dx. Calculate the integral for a spherical cap for general a and b. Then determine a and b in terms of r and R. (c) Archimedes proved that the surface area of the spherical cap is equal to the area of a circle whose radius equals the (straight-line) distance from A to a point on the circumference of the cap. Verify Archimedes' theorem. R 20
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:1.5.36. A spherical cap is the set of points on a sphere of radius R whose distance (measured
along the sphere) is at most r from a point A on the sphere. (See Figure 1.87.)
(a) Find the circumference of the spherical cap in terms of r and R.
(b) (Calculus) Verify that the area of the spherical cap is 2n R² - 2 R² cos(r/R).
Hint: Use radians. Let x be as shown in Figure 1.87. Recall that the area of the
surface obtained by revolving y = f(x) between x = a and x = b about the x-axis
is given by fo2n f(x)√1+(f'(x))²dx. Calculate the integral for a spherical cap
for general a and b. Then determine a and b in terms of r and R.
(c) Archimedes proved that the surface area of the spherical cap is equal to the area of
a circle whose radius equals the (straight-line) distance from A to a point on the
circumference of the cap. Verify Archimedes' theorem.
R
20
Figure 1.87 A spherical cap.
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