Answer:
V≅3.48πu³
Step-by-step explanation:
we have that y=2-3x² ≡ f(x)=2-3x² , then we find the cut points with the x axis
0=2-3x² ⇒ 3x²=2 ⇒ x²=2/3 ⇒ x=±√(2/3)
To find the volume of solido V=πR²h, where R=f(x) and h=dx (according graph)
so, dV=π(2-3x²)²dx ⇒ dV=π(4-12x²+9x⁴)dx integrating on both sides
∫dV=π∫(4-12x²+9x⁴)dx =∫dV/π=∫(4-12x²+9x⁴)dx =4∫dx-12∫x²dx+9∫x⁴dx=
2(4x-4x³+(9/5)x⁵)║0-√(2/3) =2( 4(√(2/3))-4(√(2/3))³+(9/5)(√(2/3))⁵ = , finally
V≅3.48πu³
The volume V of the described solid S is 3.48[tex]\pi[/tex] and this can be determined by performing the integration and using the given data.
Given :
First, determine the x-intercept by substituting (y = 0) in the given function.
[tex]\rm 0= 2-3x^2[/tex]
[tex]x^2=\dfrac{2}{3}[/tex]
[tex]x=\pm\sqrt{\dfrac{2}{3}}[/tex]
Now, the volume of solid S is given by:
[tex]\rm V = \pi r^2 h[/tex]
where r = y and h = dx.
[tex]\rm dV = \pi (2-3x^2)^2dx[/tex]
[tex]\rm dV = \pi (9x^4-12x^2+4)dx[/tex]
Now, integrate the above expression.
[tex]\rm \int dV = \int^{\sqrt{2/3} }_0 \pi (9x^4-12x^2+4)dx[/tex]
[tex]\rm V =\pi \left[ (9\dfrac{x^5}{5}-\dfrac{12x^3}{3}+4x)\right]^{\sqrt{2/3} }_0[/tex]
Simplify the above expression.
[tex]\rm V =\pi \left[ (9\dfrac{(\sqrt{\dfrac{2}{3}} )^5}{5}-\dfrac{(\sqrt{\dfrac{2}{3}} )^3}{3}+4(\sqrt{\dfrac{2}{3}} )\right][/tex]
Now, simplify the above expression by using the arithmetic operations.
[tex]\rm V \approx 3.48\pi[/tex]
For more information, refer to the link given below:
https://brainly.com/question/18990759
