\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

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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...

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Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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Chapter 13 / Lesson 10
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Cylindrical coordinates and spherical coordinates are similar to the cartesian coordinate system but applied to specifying points in a three-dimensional plane. Learn how polar coordinates apply to these calculations and their use in celestial maps.


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