• The area of a triangle is .5 times the base times the height.

• The base of this triangle is "s₁".

• So the base of this triangle is s₁ is related to the sine of angle α.

• The height is related to the cosine of angle α.

• Then the area must have a cosine of angle α times a sine of angle α.

• Check your work using the hexagon of radius = 100 feet and another way of finding the area of the triangle.

Help for other pages.

The Equation for the Planetarium Hyperbola is not a function.
To split this Hyperbola into branches which are functions gives:

and

The derivatives are respectively:

and

The second derivatives are respectively:

Both first and second derivatives are needed to find the curvature and radius of curvature along the Hyperbola. At (30.5, 0) the curvature is near 0.12860516107 and radius of a circle tangent to the vertex on the right is near 7.77573770491 feet. So a circle tangent to the vertex of this hyperbola would have an equation near:

Turn your printer to portrait to print these definite integrals.

Sky Dome Rotation for volume by the shell method (hemisphere)




Sky Dome Rotation for volume by the disk method (hemisphere)
simplifyed equals





Sky Chamber Rotation for volume by the disk method (cylinder)




Indoor Hyperbola Rotation for volume by the disk method
simplified equals





Hyperbola Rotation for volume by the disk method
simplified equals





Hyperbola Rotation For Surface Area
Circumference Times Arc Length
simplified equals




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