Guided Project 69: How big are n-balls? Topics and skills: Multiple integration We generally think of a ball as a three-dimensional sphere and the points inside of it. However, a ball may be defined in any number of dimensions and balls in n-dimensions are called n-balls, for n ≥1. For example, • the 1-ball of radius r is the interval B,(r) = {x: x ≤r) . the 2-ball of radius r is the disk B.(r) = ((x, y): (x²+y) Srl . the 3-ball of radius r is the "common ball" B,(r) = (x, y, z): (+y+2) Srl . in general, the n-ball of radius r is the set- B.(r) ((x, x Notice that for n=1, 2, and 3, the sizes (or volumes) of the n-balls are V₁(r) 2r, V(r) = x², and Va(r) 4xr13. In this project we show how to compute the volume of n-balls with n > 3. We reach the remarkable conclusion that the volumes of unit n-balls (those with r 1) do not increase as n increases. 1. Let's review the computation of the volume of the 3-ball of radius r. Working in Cartesian coordinates (even though the computation is easier in spherical coordinates), explain why I dzdydx. (The factor of 8 arises because of symmetry.) 2. Use a trigonometric substitution to evaluate the integral in Step I and confirm that V(r) = 4xr'13. 3. Here is a different way to view the calculation of Vy(r) that allows us to move to higher dimensions. We write the integral in Step I as V₁(r)-8 dx. dzdy dx Explain why the inner two integrals equal (√ 1/4 of the volume of a 2-ball of radius 4. Now we use the volume formula for a 2-ball. Show that V₂ (√√r²x²)=x(r²x²). 5. Substitute for V in the integral in Step 3 to confirm again the volume formula for a 3-ball.