009B Sample Final 1, Problem 7

(a) Find the length of the curve

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\ln (\cos x),~~~0\leq x \leq \frac{\pi}{3}} .

(b) The curve

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=1-x^2,~~~0\leq x \leq 1}

is rotated about the  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} -axis. Find the area of the resulting surface.

Foundations:
1. The formula for the length  $L$  of a curve  $y=f(x)$  where  $a\leq x\leq b$  is
$L=\int _{a}^{b}{\sqrt {1+{\bigg (}{\frac {dy}{dx}}{\bigg )}^{2}}}~dx.$
2. Recall
$\int \sec x~dx=\ln |\sec(x)+\tan(x)|+C.$
3. The surface area  $S$  of a function  $y=f(x)$  rotated about the  $y$-axis is given by
$S=\int 2\pi x\,ds,$  where $ds={\sqrt {1+{\bigg (}{\frac {dy}{dx}}{\bigg )}^{2}}}~dx.$

Solution:

(a)

Step 1:
First, we calculate  ${\frac {dy}{dx}}.$
Since  $y=\ln(\cos x),$
${\frac {dy}{dx}}={\frac {1}{\cos x}}(-\sin x)=-\tan x.$
Using the formula given in the Foundations section, we have
$L=\int _{0}^{\pi /3}{\sqrt {1+(-\tan x)^{2}}}~dx.$

Step 2:
Now, we have
${\begin{array}{rcl}L&=&\displaystyle {\int _{0}^{\pi /3}{\sqrt {1+\tan ^{2}x}}~dx}\\&&\\&=&\displaystyle {\int _{0}^{\pi /3}{\sqrt {\sec ^{2}x}}~dx}\\&&\\&=&\displaystyle {\int _{0}^{\pi /3}\sec x~dx}.\\\end{array}}$

Step 3:
Finally,
${\begin{array}{rcl}L&=&\ln |\sec x+\tan x|{\bigg |}_{0}^{\frac {\pi }{3}}\\&&\\&=&\displaystyle {\ln {\bigg |}\sec {\frac {\pi }{3}}+\tan {\frac {\pi }{3}}{\bigg |}-\ln |\sec 0+\tan 0|}\\&&\\&=&\displaystyle {\ln |2+{\sqrt {3}}|-\ln |1|}\\&&\\&=&\displaystyle {\ln(2+{\sqrt {3}})}.\end{array}}$

(b)

Step 1:
We start by calculating  ${\frac {dy}{dx}}.$
Since  $y=1-x^{2},$
${\frac {dy}{dx}}=-2x.$
Using the formula given in the Foundations section, we have
$S\,=\,\int _{0}^{1}2\pi x{\sqrt {1+(-2x)^{2}}}~dx.$

Step 2:
Now, we have  $S=\int _{0}^{1}2\pi x{\sqrt {1+4x^{2}}}~dx.$
We proceed by  $u$-substitution.
Let  $u=1+4x^{2}.$
Then,   $du=8xdx$  and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{du}{8}=xdx.}
Since the integral is a definite integral, we need to change the bounds of integration.
Plugging in our values into the equation  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=1+4x^2,}   we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_1=1+4(0)^2=1}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_2=1+4(1)^2=5.}
Thus, the integral becomes
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} S& = & \displaystyle{\int_1^5 \frac{2\pi}{8} \sqrt{u}~du}\\ &&\\ & = & \displaystyle{\frac{\pi}{4} \int_1^5 u^{\frac{1}{2}}~du.} \end{array}}

Step 3:
Now, we integrate to get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{S} & = & \displaystyle{\frac{\pi}{4}\bigg(\frac{2}{3}u^{\frac{3}{2}}\bigg)\bigg|_{1}^{5}}\\ &&\\ & = & \displaystyle{\frac{\pi}{6}u^{\frac{3}{2}}\bigg|_{1}^{5}}\\ &&\\ & = & \displaystyle{\frac{\pi}{6}(5)^{\frac{3}{2}}-\frac{\pi}{6}(1)^{\frac{3}{2}}}\\ &&\\ & = & \displaystyle{\frac{\pi}{6}(5\sqrt{5}-1)}.\\ \end{array}}

Final Answer:
(a)    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln (2+\sqrt{3})}
(b)    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{6}(5\sqrt{5}-1)}

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