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kumpel [21]
2 years ago
6

Find the volume of the solid generated when the region bounded by y= x and y= 4√x is revolved about the x-axis

Mathematics
1 answer:
MariettaO [177]2 years ago
7 0

Check the picture below, that enclosed area is pretty much our "washer".

now, to get the outer radius, or "farther" radius from the axis of rotation, what I usually do is, use the f(x) - g(x) to get the area under the curve, using the axis of rotation for g(x).

\bf \stackrel{\textit{top, f(x)}}{4\sqrt{x}}~~-~~\stackrel{\textit{bottom, g(x)}}{(0)}\implies \stackrel{\textit{farthest radius}}{4\sqrt{x}-0\implies 4\sqrt{x}} \\\\\\ \stackrel{\textit{top, f(x)}}{x}~~-~~\stackrel{\textit{bottom, g(x)}}{(0)}\implies \stackrel{\textit{closest radius}}{x-0\implies x} \\\\[-0.35em] ~\dotfill\\\\ 4\sqrt{x}=x\implies \stackrel{\textit{squaring both sides}}{16x=x^2\implies }16x-x^2=0

\bf x(16-x)=0\implies x= \begin{cases} 0\\ 16 \end{cases}\qquad \qquad \impliedby \textit{these are the bounds} \\\\[-0.35em] ~\dotfill\\\\ \displaystyle\int\limits_{0}^{16}~\pi [(4\sqrt{x})^2-(x)^2]dx\implies \pi \int\limits_{0}^{16}[16x-x^2]dx\implies \pi \int\limits_{0}^{16}16x~~-~~\pi \int\limits_{0}^{16}x^2

\bf \pi \left. 16\cdot \cfrac{x^2}{2} \right]_{0}^{16}-\left. \cfrac{\pi x^3}{3} \right]_{0}^{16}\implies \left. \pi 8x^2 \cfrac{}{}\right]_{0}^{16}-\left. \cfrac{\pi x^3}{3} \right]_{0}^{16} \\\\\\ \left( \pi [2048]-\pi [0] \right)-\left(\left[ \cfrac{4096\pi }{3} \right]-[0] \right)\implies 2048\pi -\cfrac{4096\pi }{3} \implies \cfrac{2048\pi }{3}

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3 years ago
14/1 ÷ 3/8 = please show work and answer fast​
jarptica [38.1K]

Answer: 37 1/3

Step-by-step explanation:

14/1÷3/8=?

Dividing two fractions is the same as multiplying the first fraction by the reciprocal (inverse) of the second fraction.

Take the reciprocal of the second fraction by flipping the numerator and denominator and changing the operation to multiplication. Then the equation becomes

14/1 × 8/3=?

For fraction multiplication, multiply the numerators and then multiply the denominators to get

14×8/1×3 = 112/3

This fraction cannot be reduced.

The fraction

112/3

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112÷3

Convert to a mixed number using

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112/3=37 1/3

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14/1 ÷ 3/8=37 1/3

5 0
3 years ago
If the radius of the circle is 7.2cm what is the circumference to the nearest tenth?
Aleksandr-060686 [28]

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2 years ago
24) 28 + 3k &lt; 5(–3 – 8k)<br> -4 -3<br> -2<br> -1<br> 0<br> 2<br> 3<br> 4<br> 5 6
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