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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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The quotient of a number and 7 is greater than or equal to 20
Rina8888 [55]

Answer:

a is greater than or equal to 140

Step-by-step explanation:

Multiply both sides by 7, which cancels out the 7 and turns 20 into 140

6 0
2 years ago
During a period of extreme cold, the average daily temperature dropped decreased by 112°F day. How many days did it take for the
NNADVOKAT [17]

It takes about 4 days for the temperature to decrease by 9°F.

An equation is an expression that shows the relationship between two or more variables or numbers.

Let x represent the number of days. Given that the temperature drop decreased by 1 1/2°F each day (- 1 1/2), for it to decrease by 9°F (-9):

-1 1/2 * x = -9

-1.5x = - 9

x = 4 days

It takes about 4 days for the temperature to decrease by 9°F.

5 0
2 years ago
If ray earns $120 for 15 hours of work, how much will he earn for 20 hours of work at the same rate
ss7ja [257]

Find his rate:

120 / 15 = $8 per hour

Multiply the rate by hours:

$8 x 20 hours = $160

He will earn $160

6 0
3 years ago
Find all zeros of ƒ(x) = 2x^4 – 72x^2. Then determine the multiplicity at each zero. State whether the graph will touch or cross
AleksandrR [38]

Answer: See below

Step-by-step explanation:

<u>To find the factor,</u>

f(x) = 0

\begin{aligned}&2 x^{4}-72 x^{2}=0 \\&2 x^{2}\left(x^{2}-36\right)=0 \\&x^{2}\left(x^{2}-36\right)=0 \\&x \times x \times(x-6) \times(x+6)=0 \\&\mathrm{x}=0,0,6,-6\end{aligned}

<u>So,</u>

x = 0 multiplicity 2

x = -6 multiplicity 1

x = 6 multiplicity 1

For an even multiplicty, the graph touches the x-axis, and for an odd multiplicty, the graph crosses the x-axis

<u>Therefore,</u>

x = 0 multiplicty 2, Touch

x = -6 multiplicty 1, Cross

x = 6 multiplicty 1, Cross

7 0
2 years ago
What is the area of the acute triangle given below?<br> A.18<br> B.72<br> C.36<br> D.26
Temka [501]

Hi there!

Area of a triangle: 1/2bh

Substitute 6 for h and 12 for b.

1/2(12)(6)

Multiply (12)(6).

1/2(72)

Divide 72 by 2.

36

C) 36 sq. units

Hope this helps!
3 0
3 years ago
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