For this case, we use
the cylindrical coordinates:
x² + y² = r²
dV = r dz dr dθ
The limits are:
z = 0 to z = 1/(r²)^10 = 1/r^20
r = 1 to ∞
θ = 0 to 2π
Integrating over the limits:
V = ∫ [0 to 2π] ∫ [1 to ∞] ∫ [0 to 1/r^20] r dz dr dθ
V = ∫ [0 to 2π] ∫ [1 to ∞] rz | [z = 0 to 1/r^20] dr dθ
V = ∫ [0 to 2π] ∫ [1 to ∞] 1/r^19 dr dθ
V = ∫ [0 to 2π] −1/(18r^18) |[1 to ∞] dθ
V = ∫ [0 to 2π] 1/18 dθ
V = θ/18 |[0 to 2π]
V = π/9
The volume of the volcano is an illustration of definite integral
The volume of the volcano is: [tex]\mathbf{\frac{1}{9}\pi}[/tex]
The graphs are given as:
[tex]\mathbf{z = 0}[/tex] and [tex]\mathbf{z = \frac{1}{(x^2 + y^2)^{10}}}[/tex]
The cylinder is:
[tex]\mathbf{x + y =1}[/tex]
For cylindrical coordinates, we have:
[tex]\mathbf{r^2 =x^2 + y^2}[/tex]
So, we have:
[tex]\mathbf{z = \frac{1}{(r^2)^{10}}}[/tex]
[tex]\mathbf{z = \frac{1}{r^{20}}}[/tex]
Where:
[tex]\mathbf{r = 1 \to \infty}[/tex]
[tex]\mathbf{\theta = 0 \to 2\pi}[/tex]
So, the integral is:
[tex]\mathbf{V = \int\limits^{2\pi}_0 {\int\limits^{\infty}_1 {\frac{1}{r^{20}}} \, r\ dr } \, d\theta }[/tex]
Cancel out r
[tex]\mathbf{V = \int\limits^{2\pi}_0 {\int\limits^{\infty}_1 {\frac{1}{r^{19}}} \, dr } \, d\theta }[/tex]
Rewrite as:
[tex]\mathbf{V = \int\limits^{2\pi}_0 {\int\limits^{\infty}_1 {r^{-19}} \, dr } \, d\theta }[/tex]
Integrate
[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}r^{-18}}} |\limits^{\infty}_1 \, d\theta }[/tex]
Expand
[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}(\infty^{-18} -1^{-18}) }} , d\theta }[/tex]
[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}(0 -1) }} , d\theta }[/tex]
[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}( -1) }} , d\theta }[/tex]
[tex]\mathbf{V = \int\limits^{2\pi}_0 {{\frac{1}{18} }} , d\theta }[/tex]
Integrate
[tex]\mathbf{V = \frac{1}{18}(\theta)|\limits^{2\pi}_0}[/tex]
Expand
[tex]\mathbf{V = \frac{1}{18}(2\pi - 0)}[/tex]
[tex]\mathbf{V = \frac{1}{18}(2\pi )}[/tex]
Cancel out 2
[tex]\mathbf{V = \frac{1}{9}\pi}[/tex]
Hence, the volume is: [tex]\mathbf{\frac{1}{9}\pi}[/tex]
Read more about volumes at:
https://brainly.com/question/9897385
