a) Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.
xy = 2, x = 0, y = 2, y = 4
b) Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
y = x3, y = 8, x = 0; about x = 9
Answers
(a) See the attached sketch. Each shell will have a radius y chosen from the interval [2, 4], a height of x = 2/y, and thickness ∆y. For infinitely many shells, we have ∆y converging to 0, and each super-thin shell contributes an infinitesimal volume of
2π (radius)² (height) = 4πy
Then the volume of the solid is obtained by integrating over [2, 4]:
[tex]\displaystyle 4\pi \int_2^4 y\,\mathrm dy = 2\pi y^2\bigg|_{y=2}^{y=4} = 2\pi (4^2-2^2) = \boxed{24\pi}[/tex]
(b) See the other attached sketch. (The text is a bit cluttered, but hopefully you'll understand what is drawn.) Each shell has a radius 9 - x (this is the distance between a given x value in the orange shaded region to the axis of revolution) and a height of 8 - x ³ (and this is the distance between the line y = 8 and the curve y = x ³). Then each shell has a volume of
2π (9 - x)² (8 - x ³) = 2π (648 - 144x + 8x ² - 81x ³ + 18x ⁴ - x ⁵)
so that the overall volume of the solid would be
[tex]\displaystyle 2\pi \int_0^2 (648-144x+8x^2-81x^3+18x^4-x^5)\,\mathrm dx = \boxed{\frac{24296\pi}{15}}[/tex]
I leave the details of integrating to you.
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Answer:
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Step-by-step explanation:
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Answer:
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Answers
Answer:
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Step-by-step explanation:
I'll take it as your "h" was a typo and u want to find n
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Answer:
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Step-by-step explanation:
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Answer:
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Answer:
Area= 12.56 inches²Step-by-step explanation:
To find the area of a circle: A= πr²A= 3.14 × 2²A= 3.14 × 4A= 12.56 inches²[tex]\tt{ \green{P} \orange{s} \red{y} \blue{x} \pink{c} \purple{h} \green{i} e}[/tex]
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The formula for the area of a circle is:
[tex]\sf{A=\pi r^2}}[/tex]
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Diagram:
[tex]\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\qbezier(2.3,0)(2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,-2.121)(0,-2.3)\qbezier(2.3,0)(2.121,-2.121)(-0,-2.3)\put(0,0){\line(1,0){2.3}}\put(0.5,0.3){\bf\large 2\ in}\end{picture}[/tex]
Plug in the data:
[tex]\sf{A = 3.14 \times 2^2}[/tex]
[tex]\sf{A = 3.14 \times 4}[/tex]
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Answers
Answer:
rational
Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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Answer:
The discount is 5.82
Step-by-step explanation:
To find the amount of the discount
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Find (fog)(x) and (gof)(x) f(x) = 8x-5, g(x) = 10 - 2x
help please
Answers
Answer:
[tex] (fog)(x) = f( \: g(x) \: ) \\ = 8(10 - 2x) - 5 \\ = 80 - 16x - 5 \\ = - 16x + 75[/tex]
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I hope I helped you^_^
