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3 years ago

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Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y=x−1 y=0 x=2 and x=5

about the x-axis.

1 answer:

Irina-Kira [14]3 years ago

4 0

Using disks ...
$V= \pi \int\limits^5_2 {(x-1)^{2}} \, dx =\pi(\frac{1}{3}(5^{3}-2^{3})-(5^{2}-2^{2})+(5-2))=21\pi$

The volume is 21π units³ ≈ 65.97 units³

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A Norman window has the shape of a rectangle surmounted by a semicircle. Suppose the outer perimeter of such a window must

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The base length that will maximize the area for such a window is 168.03 cm. The exact largest value of x when this occurs is 233.39 cm

Suppose we make an assumption that:

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Also, provided that the diameter of the semi-circle appears to be the base of the rectangle, then;

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and, the perimeter of the window can now be expressed as:

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Given that the perimeter = 600 cm

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$\mathbf{ \Big ( 1 + \dfrac{\pi}{2}\Big) x + 2h= 600}$

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Since h > 0, then:

$\mathbf{ h = 300 - \Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) x>0}$

By rearrangement and using the inverse rule:

$\mathbf{ x< \dfrac{ 300}{\Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) } }$

$\mathbf{ x= \dfrac{ 1200}{\Big( 2 +\pi \Big) } }$

$\mathbf{ x= 233.39 \ cm }$

Thus, the largest length x = 233.39 cm

However, the area of the window is given as:

$\mathbf{A(x) = xh + \dfrac{1}{2} \pi r^2}$

$\mathbf{A = x \Big [ 300 - \Big ( \dfrac{1}{2}+\dfrac{1}{4} \Big) x \Big ] +\dfrac{1}{2}\pi \Big(\dfrac{x}{2} \Big )^2}$

$\mathbf{A (x) = 300x - \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ cm^2}$

Now, at maximum, when the area A = 0. Taking the differentiation, we have:

$\mathbf{\dfrac{d}{dx} 300x - \dfrac{d}{dx} \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ =0}$

$\mathbf{ 300 - 2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) \ =0}$

Making x the subject of the formula, we have:

$\mathbf{x = \dfrac{1200}{4 +\pi}}$

x = 168.03 cm

Taking the second derivative:

$\mathbf{\dfrac{d}{dx} \Big [300 -2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) \Big]}$

$\mathbf{= -2 \Big( \dfrac{1}{2}+\dfrac{\pi}{8}\Big )$

Therefore, we can conclude that the maximum area that exists for such a window is 168.03 cm

Learn more about derivative here:

brainly.com/question/9964510?referrer=searchResults

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Answer:

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Step-by-step explanation:

Given

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