Answer to Question #228389 in Calculus for Unknown346307

Question #228389

Use a triple integral to determine the volume of the region bounded by z =


p


x


2 + y


2 and z = x


2 + y


2


In 1st octant


1
Expert's answer
2021-09-21T18:12:08-0400

Answer:-

To convert from cylindrical to rectangular coordinates, we use the equations



"x=r\\cos \\theta, y=r\\sin \\theta, z=z""z=\\sqrt{x^2+y^2}, z=\\sqrt{(r\\cos \\theta)^2+(r\\sin \\theta)^2}=r""z=x^2+y^2, z=(r\\cos \\theta)^2+(r\\sin \\theta)^2=r^2""r=r^2, r_1=0, r_2=1""V=\\displaystyle\\int_{0}^{\\pi\/2}d\\theta\\displaystyle\\int_{0}^{1}rdr\\displaystyle\\int_{r^2}^{r}dz""=\\displaystyle\\int_{0}^{\\pi\/2}d\\theta\\displaystyle\\int_{0}^{1}(r^2-r^3)dr""=\\dfrac{\\pi}{2}(\\dfrac{1}{3}-\\dfrac{1}{4})=\\dfrac{\\pi}{24}({units}^2)"

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