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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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dybincka [34]

Answer:

a) s = -16t^{2} + 192t + 96

b) Height of the football after 2 seconds is 448 feet.

ii. The football would not hit the ground after 2 seconds.

Step-by-step explanation:

Given:

s = -16t^{2} + V_{o}t + s_{o}

where: s is the height of the football above the ground, t is the time, V_{o} is the initial velocity and s_{o} is the initial height.

The height of the building = V_{o} = 192 feet and the initial speed of the ball = s_{o} = 96 feet per second, then;

s = -16t^{2} + 192t + 96

a) The polynomial is given as:

s = -16t^{2} + 192t + 96

b) The height of the football when t = 2 seconds is;

s = -16(2) + 192(2) + 96

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s = 448

The height of the football when t = 2 seconds is 448 feet.

ii. The football would not hit the ground after 2 seconds.

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