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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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5 0
3 years ago
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*URGENT ALGEBRA 2* Anyone know these answers? Will award brainliest.
stiv31 [10]
Correct Answer:
Second option

Solution:
The given sequence is:

2.2, 6.6, 11, ...

If we observe the sequence it is an Arithmetic Sequence as the difference between two consecutive terms is the same.

6.6 - 2.2 = 4.4 
11 - 6.6 = 4.4

First we find the general term for this sequence. 

a(n)= a_{1}+(n-1)d

a_{1}=first term = 2.2
d = 4.4

So,

a(n) = 2.2 +(n-1)*4.4 \\  \\ &#10;a(n) = 2.2 + 4.4n - 4.4 \\  \\ &#10;a(n) = 4.4n - 2.2

This general term will be placed inside the summation sign with limits from 1 to 5 to give the sum of first 5 terms of the given sequence. So the correct answer is Second Option
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2 years ago
Suppose the radius of a circle is 5.what is its diameter
Gnom [1K]
The diameter would be twice 5, or 10
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2 years ago
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Help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Oksanka [162]

Answer:

Circumfrence

Step-by-step explanation:

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