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Jet001 [13]
3 years ago
14

Simplify (reduce) 8/30 it is a fraction

Mathematics
1 answer:
defon3 years ago
3 0
4/15 is it's lowest terms. 2 is the lowest number that goes into both 8 and 30. It goes into 8 four times, and 30 15 times.
~Silver
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Multiple-Choice Integration, Picture Included, Please Include Work
ValentinkaMS [17]
\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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kirza4 [7]

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