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Problem 4 "Volume" of a four dimensional ball A circle or radius a can be expressed as x2 + y2-a2, with area value Ïα2. A sphere of radius a can be expressed as x2 +y2 + 2 = a2, with volume value-Ïαλ We can generalize the notion of the sphere to higher dimensions. Define a sphere of radius a in the 4 4 dimensional space as It is nearly impossible to visualize this object since it exists in the 4 dimensional space. But it can be accepted naturally that it should be a closed "solid" and has a similar notion of "volume" Your tasks: (1) Set up the "volume" of the ball of radius a in four dimensional space as a triple integral. (2) Calculate the triple integral you obtained in part (1) to find a volume formula for four dimensional balls. (Hint: use spherical coordinates for the (x,y,z) in this triple integral.) The answer of the volume formula is V -Ï2a4. See wikipedia page: Volume of an n-ball Hint: You may use the integral formula ofuVa2-u2du = u (2u2-02) vazu2+ a arcsin -c. 8 a