21 and 1 because both numbers are odd
First, we have to convert our function (of x) into a function of y (we revolve the curve around the y-axis). So:
![y=100-x^2\\\\x^2=100-y\qquad\bold{(1)}\\\\\boxed{x=\sqrt{100-y}}\qquad\bold{(2)} \\\\\\0\leq x\leq10\\\\y=100-0^2=100\qquad\wedge\qquad y=100-10^2=100-100=0\\\\\boxed{0\leq y\leq100}](https://tex.z-dn.net/?f=y%3D100-x%5E2%5C%5C%5C%5Cx%5E2%3D100-y%5Cqquad%5Cbold%7B%281%29%7D%5C%5C%5C%5C%5Cboxed%7Bx%3D%5Csqrt%7B100-y%7D%7D%5Cqquad%5Cbold%7B%282%29%7D%20%5C%5C%5C%5C%5C%5C0%5Cleq%20x%5Cleq10%5C%5C%5C%5Cy%3D100-0%5E2%3D100%5Cqquad%5Cwedge%5Cqquad%20y%3D100-10%5E2%3D100-100%3D0%5C%5C%5C%5C%5Cboxed%7B0%5Cleq%20y%5Cleq100%7D)
And the derivative of x:
![x'=\left(\sqrt{100-y}\right)'=\Big((100-y)^\frac{1}{2}\Big)'=\dfrac{1}{2}(100-y)^{-\frac{1}{2}}\cdot(100-y)'=\\\\\\=\dfrac{1}{2\sqrt{100-y}}\cdot(-1)=\boxed{-\dfrac{1}{2\sqrt{100-y}}}\qquad\bold{(3)}](https://tex.z-dn.net/?f=x%27%3D%5Cleft%28%5Csqrt%7B100-y%7D%5Cright%29%27%3D%5CBig%28%28100-y%29%5E%5Cfrac%7B1%7D%7B2%7D%5CBig%29%27%3D%5Cdfrac%7B1%7D%7B2%7D%28100-y%29%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%5Ccdot%28100-y%29%27%3D%5C%5C%5C%5C%5C%5C%3D%5Cdfrac%7B1%7D%7B2%5Csqrt%7B100-y%7D%7D%5Ccdot%28-1%29%3D%5Cboxed%7B-%5Cdfrac%7B1%7D%7B2%5Csqrt%7B100-y%7D%7D%7D%5Cqquad%5Cbold%7B%283%29%7D)
Now, we can calculate the area of the surface:
![A=2\pi\int\limits_0^{100}\sqrt{100-y}\sqrt{1+\left(-\dfrac{1}{2\sqrt{100-y}}\right)^2}\,\,dy=\\\\\\= 2\pi\int\limits_0^{100}\sqrt{100-y}\sqrt{1+\dfrac{1}{4(100-y)}}\,\,dy=(\star)](https://tex.z-dn.net/?f=A%3D2%5Cpi%5Cint%5Climits_0%5E%7B100%7D%5Csqrt%7B100-y%7D%5Csqrt%7B1%2B%5Cleft%28-%5Cdfrac%7B1%7D%7B2%5Csqrt%7B100-y%7D%7D%5Cright%29%5E2%7D%5C%2C%5C%2Cdy%3D%5C%5C%5C%5C%5C%5C%3D%202%5Cpi%5Cint%5Climits_0%5E%7B100%7D%5Csqrt%7B100-y%7D%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4%28100-y%29%7D%7D%5C%2C%5C%2Cdy%3D%28%5Cstar%29)
We could calculate this integral (not very hard, but long), or use
(1),
(2) and
(3) to get:
![(\star)=2\pi\int\limits_0^{100}1\cdot\sqrt{100-y}\sqrt{1+\dfrac{1}{4(100-y)}}\,\,dy=\left|\begin{array}{c}1=\dfrac{-2\sqrt{100-y}}{-2\sqrt{100-y}}\end{array}\right|= \\\\\\= 2\pi\int\limits_0^{100}\dfrac{-2\sqrt{100-y}}{-2\sqrt{100-y}}\cdot\sqrt{100-y}\cdot\sqrt{1+\dfrac{1}{4(100-y)}}\,\,dy=\\\\\\ 2\pi\int\limits_0^{100}-2\sqrt{100-y}\cdot\sqrt{100-y}\cdot\sqrt{1+\dfrac{1}{4(100-y)}}\cdot\dfrac{dy}{-2\sqrt{100-y}}=\\\\\\](https://tex.z-dn.net/?f=%28%5Cstar%29%3D2%5Cpi%5Cint%5Climits_0%5E%7B100%7D1%5Ccdot%5Csqrt%7B100-y%7D%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4%28100-y%29%7D%7D%5C%2C%5C%2Cdy%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bc%7D1%3D%5Cdfrac%7B-2%5Csqrt%7B100-y%7D%7D%7B-2%5Csqrt%7B100-y%7D%7D%5Cend%7Barray%7D%5Cright%7C%3D%20%5C%5C%5C%5C%5C%5C%3D%202%5Cpi%5Cint%5Climits_0%5E%7B100%7D%5Cdfrac%7B-2%5Csqrt%7B100-y%7D%7D%7B-2%5Csqrt%7B100-y%7D%7D%5Ccdot%5Csqrt%7B100-y%7D%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4%28100-y%29%7D%7D%5C%2C%5C%2Cdy%3D%5C%5C%5C%5C%5C%5C%202%5Cpi%5Cint%5Climits_0%5E%7B100%7D-2%5Csqrt%7B100-y%7D%5Ccdot%5Csqrt%7B100-y%7D%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4%28100-y%29%7D%7D%5Ccdot%5Cdfrac%7Bdy%7D%7B-2%5Csqrt%7B100-y%7D%7D%3D%5C%5C%5C%5C%5C%5C)
![=2\pi\int\limits_0^{100}-2\big(100-y\big)\cdot\sqrt{1+\dfrac{1}{4(100-y)}}\cdot\left(-\dfrac{1}{2\sqrt{100-y}}\, dy\right)\stackrel{\bold{(1)}\bold{(2)}\bold{(3)}}{=}\\\\\\= \left|\begin{array}{c}x=\sqrt{100-y}\\\\x^2=100-y\\\\dx=-\dfrac{1}{2\sqrt{100-y}}\, \,dy\\\\a=0\implies a'=\sqrt{100-0}=10\\\\b=100\implies b'=\sqrt{100-100}=0\end{array}\right|=\\\\\\= 2\pi\int\limits_{10}^0-2x^2\cdot\sqrt{1+\dfrac{1}{4x^2}}\,\,dx=(\text{swap limits})=\\\\\\](https://tex.z-dn.net/?f=%3D2%5Cpi%5Cint%5Climits_0%5E%7B100%7D-2%5Cbig%28100-y%5Cbig%29%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4%28100-y%29%7D%7D%5Ccdot%5Cleft%28-%5Cdfrac%7B1%7D%7B2%5Csqrt%7B100-y%7D%7D%5C%2C%20dy%5Cright%29%5Cstackrel%7B%5Cbold%7B%281%29%7D%5Cbold%7B%282%29%7D%5Cbold%7B%283%29%7D%7D%7B%3D%7D%5C%5C%5C%5C%5C%5C%3D%20%5Cleft%7C%5Cbegin%7Barray%7D%7Bc%7Dx%3D%5Csqrt%7B100-y%7D%5C%5C%5C%5Cx%5E2%3D100-y%5C%5C%5C%5Cdx%3D-%5Cdfrac%7B1%7D%7B2%5Csqrt%7B100-y%7D%7D%5C%2C%20%5C%2Cdy%5C%5C%5C%5Ca%3D0%5Cimplies%20a%27%3D%5Csqrt%7B100-0%7D%3D10%5C%5C%5C%5Cb%3D100%5Cimplies%20b%27%3D%5Csqrt%7B100-100%7D%3D0%5Cend%7Barray%7D%5Cright%7C%3D%5C%5C%5C%5C%5C%5C%3D%202%5Cpi%5Cint%5Climits_%7B10%7D%5E0-2x%5E2%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4x%5E2%7D%7D%5C%2C%5C%2Cdx%3D%28%5Ctext%7Bswap%20limits%7D%29%3D%5C%5C%5C%5C%5C%5C)
![=2\pi\int\limits_0^{10}2x^2\cdot\sqrt{1+\dfrac{1}{4x^2}}\,\,dx= 4\pi\int\limits_0^{10}\sqrt{x^4}\cdot\sqrt{1+\dfrac{1}{4x^2}}\,\,dx=\\\\\\= 4\pi\int\limits_0^{10}\sqrt{x^4+\dfrac{x^4}{4x^2}}\,\,dx= 4\pi\int\limits_0^{10}\sqrt{x^4+\dfrac{x^2}{4}}\,\,dx=\\\\\\= 4\pi\int\limits_0^{10}\sqrt{\dfrac{x^2}{4}\left(4x^2+1\right)}\,\,dx= 4\pi\int\limits_0^{10}\dfrac{x}{2}\sqrt{4x^2+1}\,\,dx=\\\\\\=\boxed{2\pi\int\limits_0^{10}x\sqrt{4x^2+1}\,dx}](https://tex.z-dn.net/?f=%3D2%5Cpi%5Cint%5Climits_0%5E%7B10%7D2x%5E2%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4x%5E2%7D%7D%5C%2C%5C%2Cdx%3D%204%5Cpi%5Cint%5Climits_0%5E%7B10%7D%5Csqrt%7Bx%5E4%7D%5Ccdot%5Csqrt%7B1%2B%5Cdfrac%7B1%7D%7B4x%5E2%7D%7D%5C%2C%5C%2Cdx%3D%5C%5C%5C%5C%5C%5C%3D%204%5Cpi%5Cint%5Climits_0%5E%7B10%7D%5Csqrt%7Bx%5E4%2B%5Cdfrac%7Bx%5E4%7D%7B4x%5E2%7D%7D%5C%2C%5C%2Cdx%3D%204%5Cpi%5Cint%5Climits_0%5E%7B10%7D%5Csqrt%7Bx%5E4%2B%5Cdfrac%7Bx%5E2%7D%7B4%7D%7D%5C%2C%5C%2Cdx%3D%5C%5C%5C%5C%5C%5C%3D%204%5Cpi%5Cint%5Climits_0%5E%7B10%7D%5Csqrt%7B%5Cdfrac%7Bx%5E2%7D%7B4%7D%5Cleft%284x%5E2%2B1%5Cright%29%7D%5C%2C%5C%2Cdx%3D%204%5Cpi%5Cint%5Climits_0%5E%7B10%7D%5Cdfrac%7Bx%7D%7B2%7D%5Csqrt%7B4x%5E2%2B1%7D%5C%2C%5C%2Cdx%3D%5C%5C%5C%5C%5C%5C%3D%5Cboxed%7B2%5Cpi%5Cint%5Climits_0%5E%7B10%7Dx%5Csqrt%7B4x%5E2%2B1%7D%5C%2Cdx%7D%20)
Calculate indefinite integral:
![\int x\sqrt{4x^2+1}\,dx=\int\sqrt{4x^2+1}\cdot x\,dx=\left|\begin{array}{c}t=4x^2+1\\\\dt=8x\,dx\\\\\dfrac{dt}{8}=x\,dx\end{array}\right|=\int\sqrt{t}\cdot\dfrac{dt}{8}=\\\\\\=\dfrac{1}{8}\int t^\frac{1}{2}\,dt=\dfrac{1}{8}\cdot\dfrac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}=\dfrac{1}{8}\cdot\dfrac{t^\frac{3}{2}}{\frac{3}{2}}=\dfrac{2}{8\cdot3}\cdot t^\frac{3}{2}=\boxed{\dfrac{1}{12}\left(4x^2+1\right)^\frac{3}{2}}](https://tex.z-dn.net/?f=%5Cint%20x%5Csqrt%7B4x%5E2%2B1%7D%5C%2Cdx%3D%5Cint%5Csqrt%7B4x%5E2%2B1%7D%5Ccdot%20x%5C%2Cdx%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bc%7Dt%3D4x%5E2%2B1%5C%5C%5C%5Cdt%3D8x%5C%2Cdx%5C%5C%5C%5C%5Cdfrac%7Bdt%7D%7B8%7D%3Dx%5C%2Cdx%5Cend%7Barray%7D%5Cright%7C%3D%5Cint%5Csqrt%7Bt%7D%5Ccdot%5Cdfrac%7Bdt%7D%7B8%7D%3D%5C%5C%5C%5C%5C%5C%3D%5Cdfrac%7B1%7D%7B8%7D%5Cint%20t%5E%5Cfrac%7B1%7D%7B2%7D%5C%2Cdt%3D%5Cdfrac%7B1%7D%7B8%7D%5Ccdot%5Cdfrac%7Bt%5E%7B%5Cfrac%7B1%7D%7B2%7D%2B1%7D%7D%7B%5Cfrac%7B1%7D%7B2%7D%2B1%7D%3D%5Cdfrac%7B1%7D%7B8%7D%5Ccdot%5Cdfrac%7Bt%5E%5Cfrac%7B3%7D%7B2%7D%7D%7B%5Cfrac%7B3%7D%7B2%7D%7D%3D%5Cdfrac%7B2%7D%7B8%5Ccdot3%7D%5Ccdot%20t%5E%5Cfrac%7B3%7D%7B2%7D%3D%5Cboxed%7B%5Cdfrac%7B1%7D%7B12%7D%5Cleft%284x%5E2%2B1%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%7D)
And the area:
![A=2\pi\int\limits_0^{10}x\sqrt{4x^2+1}\,dx=2\pi\cdot\dfrac{1}{12}\bigg[\left(4x^2+1\right)^\frac{3}{2}\bigg]_0^{10}=\\\\\\= \dfrac{\pi}{6}\left[\big(4\cdot10^2+1\big)^\frac{3}{2}-\big(4\cdot0^2+1\big)^\frac{3}{2}\right]=\dfrac{\pi}{6}\Big(\big401^\frac{3}{2}-1^\frac{3}{2}\Big)=\boxed{\dfrac{401^\frac{3}{2}-1}{6}\pi}](https://tex.z-dn.net/?f=A%3D2%5Cpi%5Cint%5Climits_0%5E%7B10%7Dx%5Csqrt%7B4x%5E2%2B1%7D%5C%2Cdx%3D2%5Cpi%5Ccdot%5Cdfrac%7B1%7D%7B12%7D%5Cbigg%5B%5Cleft%284x%5E2%2B1%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%5Cbigg%5D_0%5E%7B10%7D%3D%5C%5C%5C%5C%5C%5C%3D%20%5Cdfrac%7B%5Cpi%7D%7B6%7D%5Cleft%5B%5Cbig%284%5Ccdot10%5E2%2B1%5Cbig%29%5E%5Cfrac%7B3%7D%7B2%7D-%5Cbig%284%5Ccdot0%5E2%2B1%5Cbig%29%5E%5Cfrac%7B3%7D%7B2%7D%5Cright%5D%3D%5Cdfrac%7B%5Cpi%7D%7B6%7D%5CBig%28%5Cbig401%5E%5Cfrac%7B3%7D%7B2%7D-1%5E%5Cfrac%7B3%7D%7B2%7D%5CBig%29%3D%5Cboxed%7B%5Cdfrac%7B401%5E%5Cfrac%7B3%7D%7B2%7D-1%7D%7B6%7D%5Cpi%7D)
Answer D.
Answer:
D
Step-by-step explanation:
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