Derivation of Formula for Total Surface Area of the Sphere by Integration

The total surface area of the sphere is four times the area of great circle. To know more about great circle, see properties of a sphere. Given the radius r of the sphere, the total surface area is
 

Derivation of Formula for Volume of the Sphere by Integration

For detailed information about sphere, see the Solid Geometry entry, The Sphere.
 

The formula for the volume of the sphere is given by

Solution to Problem 696-697 | Beam Deflection by Method of Superposition

Problem 696
In Fig. P-696, determine the value of P for which the deflection under P will be zero.
 

Solution to Problem 694-695 | Beam Deflection by Method of Superposition

Problem 694
The frame shown in Fig. P-694 is of constant cross section and is perfectly restrained at its lower end. Compute the vertical deflection caused by the couple M.
 

Inverted L Frame with Moment Load

 

Solution to Problem 693 | Beam Deflection by Method of Superposition

Problem 693
Determine the value of EIδ at the left end of the overhanging beam in Fig. P-693.
 

Solution to Problem 692 | Beam Deflection by Method of Superposition

Problem 692
Find the value of EIδ midway between the supports for the beam shown in Fig. P-692. (Hint: Combine Case No. 11 and one half of Case No. 8.)
 

Solution to Problem 691 | Beam Deflection by Method of Superposition

Problem 691
Determine the midspan deflection for the beam shown in Fig. P-691. (Hint: Apply Case No. 7 and integrate.)
 

Solution to Problem 690 | Beam Deflection by Method of Superposition

Problem 690
The beam shown in Fig. P-690 has a rectangular cross section 50 mm wide. Determine the proper depth d of the beam if the midspan deflection of the beam is not to exceed 20 mm and the flexural stress is limited to 10 MPa. Use E = 10 GPa.
 

Solution to Problem 689 | Beam Deflection by Method of Superposition

Problem 689
The beam shown in Fig. P-689 has a rectangular cross section 4 inches wide by 8 inches deep. Compute the value of P that will limit the midspan deflection to 0.5 inch. Use E = 1.5 × 106 psi.

Solution to Problem 688 | Beam Deflection by Method of Superposition

Problem 688
Determine the midspan value of EIδ at the left end of the beam shown in Fig. P-688.
 

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