Difference between revisions of "009B Sample Final 2, Problem 5"

Latest revision as of 17:28, 2 December 2017

(a) Find the area of the surface obtained by rotating the arc 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^3=x}

between $(0,0)$ 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 (1,1)} 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.

(b) Find the length of the arc

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+9x^{\frac{3}{2}}}

between the points 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 (1,10)} 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 (4,73).}

Solution

Detailed Solution

Return to Sample Exam

@@ Line 3: / Line 3: @@
 ::<math>y^3=x</math>
-<span class="exam">between <math style="vertical-align: -5px">(0,0)</math> and <math style="vertical-align: -5px">(1,1)</math> about the <math style="vertical-align: -4px">y</math>-axis.
+<span class="exam">between &nbsp;<math style="vertical-align: -5px">(0,0)</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">(1,1)</math>&nbsp; about the &nbsp;<math style="vertical-align: -4px">y</math>-axis.
 <span class="exam">(b) Find the length of the arc
@@ Line 9: / Line 9: @@
 ::<math>y=1+9x^{\frac{3}{2}}</math>
-<span class="exam">between the points <math style="vertical-align: -5px">(1,10)</math> and <math style="vertical-align: -5px">(4,73).</math>
+<span class="exam">between the points &nbsp;<math style="vertical-align: -5px">(1,10)</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">(4,73).</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009B Sample Final 2, Problem 5 Solution|'''<u>Solution</u>''']]
-|-
-|'''1.''' The formula for the length &nbsp;<math style="vertical-align: 0px">L</math>&nbsp; of a curve &nbsp;<math style="vertical-align: -5px">y=f(x)</math>&nbsp; where &nbsp;<math style="vertical-align: -3px">a\leq x \leq b</math>&nbsp; is
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp;<math>L=\int_a^b \sqrt{1+\bigg(\frac{dy}{dx}\bigg)^2}~dx.</math>
-|-
-|'''2.''' The surface area &nbsp;<math style="vertical-align: 0px">S</math>&nbsp; of a function &nbsp;<math style="vertical-align: -5px">y=f(x)</math>&nbsp; rotated about the &nbsp;<math style="vertical-align: -4px">y</math>-axis is given by
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -13px">S=\int 2\pi x\,ds,</math>&nbsp; where <math style="vertical-align: -18px">ds=\sqrt{1+\bigg(\frac{dy}{dx}\bigg)^2}.</math>
-|}
-'''Solution:'''
+[[009B Sample Final 2, Problem 5 Detailed Solution|'''<u>Detailed Solution</u>''']]
-'''(a)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|
-|-
-|
-|-
-|
-|-
-|
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|
-|-
-|
-|-
-|
-|-
-|
-|}
-'''(b)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we calculate &nbsp;<math>\frac{dy}{dx}.</math>
-|-
-|Since <math>y=1+9x^{\frac{3}{2}},</math> we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{dy}{dx}=\frac{27\sqrt{x}}{2}.</math>
-|-
-|Then, the arc length &nbsp;<math>L</math>&nbsp; of the curve is given by
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>L=\int_1^4 \sqrt{1+\bigg(\frac{27\sqrt{x}}{2}\bigg)^2}~dx.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Then, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>L=\int_1^4 \sqrt{1+\frac{27^2x}{2^2}}~dx.</math>
-|-
-|Now, we use &nbsp;<math>u</math>-substitution.
-|-
-|Let &nbsp;<math>u=1+\frac{27^2x}{2^2}.</math>
-|-
-|Then, &nbsp; <math>du=\frac{27^2}{2^2}dx</math>&nbsp; and &nbsp;<math>dx=\frac{2^2}{27^2}du.</math>
-|-
-|Also, since this is a definite integral, we need to change the bounds of integration.
-|-
-|We have &nbsp; <math>u_1=1+\frac{27^2(1)}{2^2}=1+\frac{27^2}{2^2}</math>
-|-
-|and &nbsp; <math>u_2=1+\frac{27^2(4)}{2^2}=1+27^2.</math>
-|-
-|Hence, we now have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>L=\int_{1+\frac{27^2}{2^2}}^{1+27^2} \frac{2^2}{27^2}u^{\frac{1}{2}}~du.</math>
-|-
-|
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 3: &nbsp;
-|-
-|Therefore, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{L} & = & \displaystyle{\frac{2^2}{27^2} \bigg(\frac{2}{3}u^{\frac{3}{2}}\bigg)\bigg|_{1+\frac{27^2}{2^2}}^{1+27^2}}\\
-&&\\
-& = & \displaystyle{\frac{2^3}{3^4} u^{\frac{3}{2}}\bigg|_{1+\frac{27^2}{2^2}}^{1+27^2}}\\
-&&\\
-& = & \displaystyle{\frac{2^3}{3^4} (1+27^2)^{\frac{3}{2}}-\frac{2^3}{3^4} \bigg(1+\frac{27^2}{2^2}\bigg)^{\frac{3}{2}}.}
-\end{array}</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|'''(a)'''
-|-
-|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; <math>\frac{2^3}{3^4} (1+27^2)^{\frac{3}{2}}-\frac{2^3}{3^4} \bigg(1+\frac{27^2}{2^2}\bigg)^{\frac{3}{2}}</math>
-|}
 [[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009B Sample Final 2, Problem 5"

Latest revision as of 17:28, 2 December 2017

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