Respuesta :
The volume of the region bounded by the sphere ρ=34cosϕ and the hemisphere ρ=17, z ≥ 0 is 14148.42 units³.
What is the volume?
- Volume is a measurement of three-dimensional space that is occupied.
- It is frequently numerically quantified using SI-derived units (such as the cubic meter and liter) or various imperial units (such as the gallon, quart, and cubic inch).
- The volume of a container is commonly understood to be its capacity; that is, the amount of fluid (gas or liquid) that the container can hold, rather than the amount of space that the container occupies.
To find the volume of the region bounded by the sphere ρ=34cosϕ and the hemisphere ρ=17, z ≥ 0:
- To begin, we should note that this region is ambiguous (whoever came up with it has failed us), as we don't know whether it is inside the hemisphere and outside the total sphere, or inside the hemisphere and outside the total sphere.
- So we'll interpret this in the simplest way possible, we'll bind the region above by the total sphere and the region below by the hemisphere.
- Then we have 17 ≤ ρ ≤ 34 cos ϕ.
- Bounding this region allows us to also bound ϕ below by 0.
- Then its upper bound is at the spheres' intersection.
[tex]\begin{aligned}34 \cos \phi &=17 \\\cos \phi &=\frac{1}{2} \\\phi &=\frac{\pi}{3}\end{aligned}[/tex]
There are no restrictions on the other angle so [tex]\theta \in[0,2 \pi][/tex], and we get:
[tex]\begin{aligned}V &=\int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{17}^{34 \cos \phi} \rho^{2} \sin \phi d \rho d \phi d \theta \\&=\int_{0}^{2 \pi} d \theta \int_{0}^{\pi / 3} \sin \phi\left[\frac{1}{3} \rho^{3}\right]_{17}^{34 \cos \phi} d \phi \\&=(2 \pi) \frac{1}{3} \int_{0}^{\pi / 3} \sin \phi\left(34^{3} \cos ^{3} \phi-17^{3}\right) d \phi \\&=\frac{2 \pi}{3} \int_{0}^{\pi / 3} 34^{3} \cos ^{3} \phi \sin \phi-17^{3} \sin \phi d \phi\end{aligned}[/tex]
[tex]\begin{aligned}&=\frac{2 \pi}{3}\left[-\frac{34^{3}}{4} \cos ^{4} \phi+17^{3} \cos \phi\right]_{0}^{\pi / 3} \\&=\frac{2 \pi}{3}\left[-\frac{34^{3}}{4}\left(\left(\frac{1}{2}\right)^{4}-1\right)+17^{3}\left(\frac{1}{2}-1\right)\right] \\&=\frac{54043 \pi}{12} \\&\approx 14148.42\end{aligned}[/tex]
Therefore, the volume of the region bounded by the sphere ρ=34cosϕ and the hemisphere ρ=17, z ≥ 0 is 14148.42 units³.
Know more about volumes here:
https://brainly.com/question/1972490
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The correct question is given below:
Use spherical coordinates to find the volume of the region bounded by the sphere ρ=34cosϕ and the hemisphere ρ=17, z ≥ 0.
