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Valentin [98]
3 years ago
14

Multiple-Choice Integration, Picture Included, Please Include Work

Mathematics
1 answer:
ValentinkaMS [17]3 years ago
4 0
\displaystyle\int_0^2\sqrt{4-x^2}\,\mathrm dx

Recall that a circle of radius 2 centered at the origin has equation

x^2+y^2=4\implies y=\pm\sqrt{4-x^2}

where the positive root gives the top half of the circle in the x-y plane. The definite integral corresponds to the area of the right half of this top half. Since the area of a circle with radius r is \pi r^2, it follows that the area of a quarter-circle would be \dfrac{\pi r^2}4.

You have r=2, so the definite integral is equal to \dfrac{2^2\pi}4=\pi.

Another way to verify this is to actually compute the integral. Let x=2\sin u, so that \mathrm dx=2\cos u\,\mathrm du. Now

\displaystyle\int_0^2\sqrt{4-x^2}\,\mathrm dx=\int_0^{\pi/2}\sqrt{4-(2\sin u)^2}(2\cos u)\,\mathrm du=4\int_0^{\pi/2}\cos^2u\,\mathrm du

Recall the half-angle identity for cosine:

\cos^2u=\dfrac{1+\cos2u}2

This means the integral is equivalent to

\displaystyle2\int_0^{\pi/2}(1+\cos 2u)\,\mathrm du=2u+\sin2u\bigg|_{u=0}^{u=\pi/2}=\pi
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Ivan and Tanya share £150 in the ration of 4:1
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Answer:

Money Ivan gets   extra = 120 - 30 (or 3x) = 90

Step-by-step explanation:

Ivan = 4x

Tanya = 1x

Total = 150

4x + x = 150

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Ivan = 4x = 120

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Money Ivan gets   extra = 120 - 30 (or 3x) = 90

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olchik [2.2K]

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Answer:GCF: y^4

Step-by-step explanation:

Factors of:

y^4= (y) (y) (y) (y) = y^4 (1)

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An investor has purchased 1,000 shares of stock at a price of $10 per share. The stock pays no dividends, and historically has a
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D

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Here, we want to select which of the options explains the scenario in the question.

Firstly, 1,000 shares were purchased at $10 per share.

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