#topicpath * 5次元球の体積 [#a6a7951f] ** 3次元球の体積 [#u2327daf] # # 3D # (%i10) integrate(1, y,-sqrt(r^2-x^2-y^2), sqrt(r^2-x^2-y^2)); (%i11) integrate(%,y,-sqrt(r^2-x^2),sqrt(r^2-x^2)); (%i12) integrate(%,x,-r,r); 答え… $ \frac{4\pi r^3}{3} $~ ~ ** 4次元球の体積 [#a225f74c] # # 4D # (%i13) integrate(2*sqrt(r^2-x1^2-x2^2-x3^2), x1, -sqrt(r^2-x2^2-x3^2), sqrt(r^2-x2^2-x3^2)); (%i14) integrate(%, x2, -sqrt(r^2-x3^2), sqrt(r^2-x3^2)); (%i15) integrate(%, x3, -sqrt(r^2), sqrt(r^2)); 答え… $ \frac{\pi^2 r^4}{2} $~ ~ ** 5次元球の体積 [#a0e491df] # # 5D # (%i20) integrate(2*sqrt(r^2-x1^2-x2^2-x3^2-x4^2), x1, -sqrt(r^2-x2^2-x3^2-x4^2), sqrt(r^2-x2^2-x3^2-x4^2)); (%i21) integrate(%, x2, -sqrt(r^2-x3^2-x4^2), sqrt(r^2-x3^2-x4^2)); (%i22) integrate(%, x3, -sqrt(r^2-x4^2), sqrt(r^2-x4^2)); (%i23) integrate(%, x4, -r, r); 答え… $ \frac{8\pi^2 r^{5}}{15} $~ ~