Use cylindrical coordinates to show that the volume of a sphere of radius a from which a central cylinder of radius b has been removed, where 0 < b < a, only depends on the height of the band that results. In particular, this implies that such a band of radius 2 m and height 1 m has the same volume as such a band of radius 6,400 km (the radius of the earth) and height 1 m.

Q: 41. Prove a famous result of Archimedes: The surface area of the portion of the sphere of radius R…

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Q: Find the dimensions of the cone of maximum volume inscribed in a sphere of radius 8 cm. [Note:…

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Q: A cylindrical can of height h and radius r is to be made with a fixed volume V. Show that the area…

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Q: 3]. , Let V = ar²h, where V the volume of a right circular cone of radiusr and height h. Show that…

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Q: Prove that the area of a trapezoid whose altitude has length h and whose median has length m is A =…

A: Let's consider a trapezoid ABCD with bases AD=b1   and  BC=b2 and altitudes AE and DF measures h…

Q: 日-1- Find the volume of the parallelepiped determined by w = |0,v = 1, and u =|1

A: The Volume of parallelepiped formed between three vectors A→,B→ and  C→ is given by the scalar…

Q: 12. Calculate the area and volume of a sphere knowing that the length of its equator is L = 12p cm.

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Q: 2. Consider part of the graph of y = ! for x > 1. By rotating the graph about the x-axis, the…

A: Given : The graph of y=1x for x≥1 By rotating the graph about the x-axis  we get the unbounded solid…

Q: 69. If lines are drawn parallel to the coordinate axes through a point P on the parabola y² = kx, k…

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Q: 7) An elliptic cross section of a highway culvert can be described as the region within the ellipse:…

A: Here is the step-by-step explanationPart A (Problem 7):To find the area of the ellipse given by the…

Q: When a ship is sailing, the total cost per hour includes a fixed part p and the fuel cost q that is…

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Q: 1)Find the area bounded by the given curves: a)y = 6x2 and y = 24   b)y = 3x2 − 12 and y = 0

A: The given function are y=6x2 and y=24. The area bounded by the given curves as shown below:

Q: Show the surface area and volume of a sphere with radius ris 47r² and ¹³ πι respectively.

A: The objective is to show that the surface area and volume of a sphere with radius r is 4πr2 and…

Q: A right cylinder of radius r is inscribed in a hollow sphere of fixed radius a. The

A: Given, a right cylinder of radius 'r', is inscribed in a hollow sphere of fixed radius 'a'. The…

Q: If the line x = 1- y, with 0 < y<1 is revolved about the y-axis. Find the generated lateral surface…

A: Given  the line x = 1-y with 0≤y≤1 is resolving about the y-axis we have to find the generated…