Mathematics: The base of a solid is the region in the first quadrant bounded above by the line y = 5, below by y = sin−1(x), and to the right by the line x = 1. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?

The base of a solid is the region in the first

Ответ:

Miamonroe2004

21.01.2022

Each cross section is a square with side length equal to the vertical distance between the curves y=5 and $y=\sin^{-1}x$ , with $0\le x\le1$ , or $5-\sin^{-1}x$ . Then each section has an area of $(5-\sin^{-1}x)^2$ , so the volume of this solid is

$\displaystyle\int_0^1(5-\sin^{-1}x)^2\,\mathrm dx=\boxed{33-5\pi+\frac{\pi^2}4}$

For computing the integral, consider the substitution

$u=5-\sin^{-1}x\implies x=\sin(5-u)\implies\mathrm dx=-\cos(5-u)\,\mathrm du$

Then the integral becomes

$\displaystyle\int_5^{5-\pi/2}u^2(-\cos(5-u))\,\mathrm du$

$=\displaystyle\int_{5-\pi/2}^5u^2\cos(u-5)\,\mathrm du$

Integrate by parts twice; for the first round, take

$f=u^2\implies\mathrm df=2u\,\mathrm du$

$\mathrm dg=\cos(u-5)\,\mathrm du\implies g=\sin(u-5)$

$\implies\displaystyle u^2\sin(u-5)\bigg|_{5-\pi/2}^5-2\int_{5-\pi/2}^5u\sin(u-5)\,\mathrm du$

For the second round, take

$f=u\implies\mathrm df=\mathrm du$

$\mathrm dg=-\sin(u-5)\,\mathrm du\implies g=\cos(u-5)$

$\implies\displaystyle\bigg(u^2\sin(u-5)+u\cos(u-5)\bigg)\bigg|_{5-\pi/2}^5-\int_{5-\pi/2}^5\cos(u-5)\,\mathrm du$

and the rest is trivial.

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Ответ:

pinapunapula

27.01.2021

Mathematics

See explanation

Step-by-step explanation:

The average rainfall when you add all 10 years of rainfall up and divide by 10 is an average of 59.946 inches of rain each year. The equation for the data is y = -0.53x + 64.45, this means that the rainfall is getting less each year at a -0.53 inches of rain each year.

x = year (2004 would be 4, etc)

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The base of a solid is the region in the first quadrant bounded above by the line y = 5, below by y = sin−1(x), and to the right by the line x = 1. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?

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