This discussion focuses on determining the moment of inertia of an area with respect to the y-axis using two different methods: one with thickness dx and the other with thickness dy. The relevant equations include the integral for moment of inertia, specifically ## I_x=\int _{ }^{ }y^2dA ##, and the area mass density denoted as ##\sigma##. The user explores the equivalence of two integral expressions, specifically 2\sigma ∫₀¹ x²y dx and 32σ ∫₀^{π/2} sin²θ cos²θ dθ, to derive the same moment of inertia.
PREREQUISITESStudents in physics or engineering, particularly those studying mechanics and structural analysis, as well as educators looking for examples of calculating moment of inertia using different methods.
![<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}I_x=\int_{ }^{ }y^2dA\\
I_x=\int_0^4y^22xdy\ \left[y=4-4x^2,\ \textcolor{#E94D40}{\sqrt{\frac{4-y}{4}}=x}\right]\\
I_x=2\int_0^4y^2\sqrt{\frac{4-y}{4}}dy\ \left(u=\frac{4-y}{4},\ dy=-4du\right)\\
I_x=-8\int_0^4y^2\sqrt{u}du\ \left[u=\frac{4-y}{4},\ \textcolor{#E94D40}{4-4u=y}\right]\\
I_x=-8\int_0^4\left(4-4u\right)^2\sqrt{u}du\\
I_x=-8\int_0^4\left(4^2-2\cdot4\cdot4u+\left(4u\right)^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16-32u+16u^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16\sqrt{u}-32u\sqrt{u}+16u^2\sqrt{u}\right)du\\
I_x=-\frac{32\cdot8}{3}u^{\frac{3}{2}}+\frac{64\cdot8}{5}u^{\frac{5}{2}}-\frac{32\cdot8}{7}u^{\frac{7}{2}}\ \left[\textcolor{#E94D40}{u=\frac{4-y}{4}}\right]\\
I_x=-\frac{32\cdot8}{3}\left(\frac{4-y}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-y}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-y}{4}\right)^{\frac{7}{2}}\begin{bmatrix}4\\
0\end{bmatrix}\\
I_x=\cancel{\textcolor{#E94D40}{-\frac{32\cdot8}{3}\left(\frac{4-4}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-4}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-4}{4}\right)^{\frac{7}{2}}}}\\
-\left(-\frac{32\cdot8}{3}\left(\frac{4-0}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-0}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-0}{4}\right)^{\frac{7}{2}}\right)\\
I_x=19.5\mathit{\text{pulg}}^4\\
\ \end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mrow data-mjx-texclass=ORD/><mrow data-mjx-texclass=ORD/></msubsup><msup><mi>y</mi><mn>2</mn></msup><mi>d</mi><mi>A</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><mn>2</mn><mi>x</mi><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mo>=</mo><mi>x</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mn>2</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mi>d</mi><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>d</mi><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo>=</mo><mi>y</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><msup><mn>4</mn><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>4</mn><mi>u</mi><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>16</mn><mo>−</mo><mn>32</mn><mi>u</mi><mo>+</mo><mn>16</mn><msup><mi>u</mi><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr](https://lh6.googleusercontent.com/w2X_A8frcymArTvwcFv7xwyy1UuPgDt84Kue-NahCFeqIzXBqslUlQUwgsYBbFxP2rcFLIHc1DxyfUiKz6SHshk0V304Zs37Ja-ZcnMUqzQLvw_CsaQl-f8S6uODGM3g2vUCpRFC=s1600 "<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}I_x=\int_{ }^{ }y^2dA\\
I_x=\int_0^4y^22xdy\ \left[y=4-4x^2,\ \textcolor{#E94D40}{\sqrt{\frac{4-y}{4}}=x}\right]\\
I_x=2\int_0^4y^2\sqrt{\frac{4-y}{4}}dy\ \left(u=\frac{4-y}{4},\ dy=-4du\right)\\
I_x=-8\int_0^4y^2\sqrt{u}du\ \left[u=\frac{4-y}{4},\ \textcolor{#E94D40}{4-4u=y}\right]\\
I_x=-8\int_0^4\left(4-4u\right)^2\sqrt{u}du\\
I_x=-8\int_0^4\left(4^2-2\cdot4\cdot4u+\left(4u\right)^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16-32u+16u^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16\sqrt{u}-32u\sqrt{u}+16u^2\sqrt{u}\right)du\\
I_x=-\frac{32\cdot8}{3}u^{\frac{3}{2}}+\frac{64\cdot8}{5}u^{\frac{5}{2}}-\frac{32\cdot8}{7}u^{\frac{7}{2}}\ \left[\textcolor{#E94D40}{u=\frac{4-y}{4}}\right]\\
I_x=-\frac{32\cdot8}{3}\left(\frac{4-y}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-y}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-y}{4}\right)^{\frac{7}{2}}\begin{bmatrix}4\\
0\end{bmatrix}\\
I_x=\cancel{\textcolor{#E94D40}{-\frac{32\cdot8}{3}\left(\frac{4-4}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-4}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-4}{4}\right)^{\frac{7}{2}}}}\\
-\left(-\frac{32\cdot8}{3}\left(\frac{4-0}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-0}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-0}{4}\right)^{\frac{7}{2}}\right)\\
I_x=19.5\mathit{\text{pulg}}^4\\
\ \end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mrow data-mjx-texclass=ORD/><mrow data-mjx-texclass=ORD/></msubsup><msup><mi>y</mi><mn>2</mn></msup><mi>d</mi><mi>A</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><mn>2</mn><mi>x</mi><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mo>=</mo><mi>x</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mn>2</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mi>d</mi><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>d</mi><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo>=</mo><mi>y</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><msup><mn>4</mn><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>4</mn><mi>u</mi><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>16</mn><mo>−</mo><mn>32</mn><mi>u</mi><mo>+</mo><mn>16</mn><msup><mi>u</mi><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr")
waat, Can you tell me the name of this method?anuttarasammyak said:
This relevant equation seems inappropriate because x should be squared for y-axis rotation inertia.Tapias5000 said: