Question

Goal: Use integration to derive the volume of the solid sphere in dimensions above 3 (R4, Rʻ,...). Notation & Terminology: Use V, and S, for the "volume" and "surface area" of an n- dimensional solid sphere. Thus "Volume" is not always in cubic units, it is in units^n. So, similarly “surface area" is in units (n-1) and is the measurement of the boundary. 1. Look up & become familiar with the formulas for V, and S. Start in R', what is a 1-dimensional "sphere"? Draw an appropriate picture and use your formulas for V, and S. 2. Use integration to derive (prove) the formulas for V, and S. Notes: The 2-D "volume" is really the regular area inside of x + y2 = R. Similarly, 2-D "surface area" is really length. Feel free to only integrate over a portion of the disk and use an appropriate multiplier. There are at least 4 valid integration approaches for V Tone integral using Calc 2 methods, one integral using polar coordinates, a double integral, ...). Use any valid approach and INCLUDE ALL DETAILS! See section 6.5 for a possible approach for Sz. Notice that S = V. 3. Use integration to derive the formulas for V, and Sg. [See section 6.6 for one possible technique for S,. Again, there are multiple valid approaches for V, Notice that S = V.] 4. Extend the above ideas to derive the formula for V4, the "volume" contained inside of x2 + y2 +2? +W2 = R. Please show all details - this may take two (or more) pages of work! Without proof, use SE V to find S. 5. Derive the formula for Vs. Feel free to use tables to help with any integration. Without proof, use S V , to find Ss.

Answer 1

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