"Obtain the volume of the solid which is..." was answered by our experts

ruigE

Answered question

2020-11-27

Obtain the volume of the solid which is bounded by a circular paraboloid

z = x^{2} + y^{2}

, cylinder

x^{2} + y^{2} = 4

, and Coordinate plane. And the solid is in the

(x \geq 0, y \geq 0, z \geq 0)

Faiza Fuller

Skilled2020-11-28Added 108 answers

Cylindrical coordinates can be given as,

x = r \cos θ

y = r \sin θ

z = z

x^{2} + y^{2} = r^{2}

Now the r will be 2 as

x^{2} + y 2 = 2^{2}

. Limits of region can be given as,

R {(r, θ, z) : 0 \leq r \leq 2, 0 \leq θ \leq \frac{π}{2}, 0 \leq z \leq r^{2}}

Volume of solid in cylindrical coordinate plane is given as,

V = \oint_{R} r d r d θ d z

Calculate the volume using the values,

V = \int_{0}^{\frac{π}{2}} \int_{0}^{2} \int_{0}^{r^{2}} r d z d r d θ

= \int_{0}^{\frac{π}{2}} \int_{0}^{2} r {[z]}_{0}^{r^{2}} d r d θ

= \int_{0}^{\frac{π}{2}} \int_{0}^{2} r^{3} d r d θ

= \int_{0}^{\frac{π}{2}} \int_{0}^{2} {[\frac{r^{4}}{4}]}_{0}^{2} d θ

= \int_{0}^{\frac{π}{2}} 4 d θ

= 4 {[t \hat{e}]}_{0}^{\frac{π}{2}}

= 4 [\frac{π}{2} - 0]

= 2 π

Volume of the given solid is

2 π

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