1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
aksik [14]
3 years ago
7

99 POINT QUESTION, PLUS BRAINLIEST!!!

Mathematics
2 answers:
VladimirAG [237]3 years ago
6 0
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}

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)}

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)

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}}=\\\\\\

=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})=\\\\\\

=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}

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}}

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}

Answer D.
ra1l [238]3 years ago
6 0
Its A, you can use the pictures to show you

You might be interested in
Which graph shows the solution to this system of inequalities?<br> y &lt; 3x - 4<br> y ≥ x + 1
timama [110]

Answer:

B

Step-by-step explanation:

8 0
2 years ago
The Pythagorean Theorem can only be used with which type of triangle?
valkas [14]
Right triangle because  a2<span> + b</span>2<span> = c</span><span>2</span>
4 0
3 years ago
What is the area of the shaded region of the figure
Alborosie
So we're looking at two rectangles, one cut out of the other, so all you do is find the area of the big one, base times height equals area, minus the area of the small one, base times height equals area. So the equation you have to solve for is this,

Area=(12.6X14)-(3X8.4)
6 0
2 years ago
Read 2 more answers
Graph y = tanx for -pi/4 ≤ x ≤ pi/4. What is the range?
Debora [2.8K]

Answer:

( -1, 1 )

Step-by-step explanation:

For f ( x ) = tan x ;   Range = R (real no.)

Range in interval = ( -1, 1 )

3 0
3 years ago
Read 2 more answers
Simplify problem if possible
sergeinik [125]
\left(\frac{(x^4y^{-2})^4}{(2x^2)^2\cdot2x^0y^3}\right)^2=\\&#10;\left(\frac{x^{16}y^{-8}}{4x^4\cdot2y^3}\right)^2=\\&#10;\left(\frac{x^{16}y^{-8}}{8x^4y^3}\right)^2=\\&#10;\left(\frac{x^{12}y^{-11}}{8}\right)^2=\\&#10;\frac{x^{24}y^{-22}}{64}
3 0
3 years ago
Other questions:
  • Can you please help me answer this question <br> Negative n=7+6n
    5·1 answer
  • A midpoint of sequence of numbers. <br> A. Mean <br> B. Median <br> C. Mode <br> D. Zero
    14·2 answers
  • An "a" value of less than 1 produces a graph with an exponential decay . True or false ?
    8·1 answer
  • Sis)<br> What is the area of the regular hexagon?
    12·1 answer
  • 0.8x+3.2-4.3x=7.7-1.2x
    15·2 answers
  • What does x = when 2x + 6 = 6x-2
    10·2 answers
  • -6y + 12 = - 66<br> I feel like I lost my brain today I can’t think right... can you help me out
    15·2 answers
  • Х<br> A number decreased by 4 is greater than 13.
    14·1 answer
  • 6. Find the slope of the line that passes through the points: (4, 5) and (-2, 8) Vour answer​
    6·2 answers
  • -6 + 6 how do you solve it
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!