Answer to Question #228389 in Calculus for Unknown346307

Question

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

Answer 1

Answer:-

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

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