\int_{0}^{\pi }\int_{0}^{3}\int_{0}^{r^{2}}rdzdrd\theta A) Describe the solid determined by the...

Question:

{eq}\int_{0}^{\pi }\int_{0}^{3}\int_{0}^{r^{2}}rdzdrd\theta {/eq}

A) Describe the solid determined by the region of integration. (Include a sketch)

B) Evaluate the integral to find the volume of the solid.

Using Cylindrical Coordinate System To Find Volume:


The volume of any solid bounded by {eq}\displaystyle z=z_1(x,y) {/eq} and {eq}\displaystyle z=z_2(x,y) {/eq} and {eq}\displaystyle y=y_1 {/eq} and {eq}\displaystyle y=y_2 {/eq} and {eq}\displaystyle x=x_1 {/eq} and {eq}\displaystyle x_2 {/eq} can be calculated by setting up a triple integral using cylindrical coordinate systems as follows,

$$\displaystyle \text{ Volume }=\int_{x_1}^{x_2}\int_{y_1}^{y_2}\int_{z_1}^{z_2} \ dz \ dy \ dx $$. If the surfaces and curves bounding the volume are consisting of equations elated to a circle i.e. {eq}\displaystyle x^2+y^2=r^2 {/eq} then the integral may ne converted to cylindrical coordinates by using the substitution, {eq}\displaystyle x=r\cos(\theta) {/eq} and {eq}\displaystyle y=r\sin(\theta) {/eq}. this will convert the given integral to,

$$\displaystyle \text{ Volume }=\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}\int_{z_1}^{z_2} \ r \ dz \ dy \ dx $$


Answer and Explanation: 1

Become a Study.com member to unlock this answer!

View this answer


A)

The given integral for volume defines a paraboloid with its apex point at the origin and only the half along the right of the Y-Z plane is...

See full answer below.


Learn more about this topic:

Loading...
Cylindrical & Spherical Coordinates: Definition, Equations & Examples

from

Chapter 13 / Lesson 10
23K

Learn how to convert between Cartesian, cylindrical and spherical coordinates. Discover the utility of representing points in cylindrical and spherical coordinates.


Related to this Question

Explore our homework questions and answers library