Bài 3: Ứng dụng của tích phân trong hình học
Hướng dẫn giải Bài 3 (Trang 121 SGK Toán Giải tích 12)
<p><strong>Bài 3 (Trang 121 SGK Toán Giải tích 12):</strong></p>
<p>Parapol <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></math> chia hình tròn có tâm tại gốc tọa độ, bán kính <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt></math> thành hai phần. Tìm tỉ số diện tích của chúng.</p>
<p><strong><em>Hướng dẫn giải:</em></strong></p>
<p>Phương trình đường tròn tâm O bán kính <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt></math> là: (C): <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn></math>. Tung độ giao điểm của (C) và (P) là:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>8</mn><mo>=</mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">y</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>4</mn><mo> </mo><mo>(</mo><mi>loại</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn></mtd></mtr></mtable></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><mo> </mo><mo>⇔</mo><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>±</mo><mn>2</mn></math></p>
<p>Gọi S<sub>1</sub> là diện tích giới hạn bởi (C) và (P) ở phía trên trục hoành.</p>
<p>Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">S</mi><mn>1</mn></msub><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><msubsup><mo>∫</mo><mn>0</mn><mn>2</mn></msubsup><mo>(</mo><msqrt><mn>8</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt><mo>-</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi></mfrac><mo>)</mo><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><msubsup><mo>∫</mo><mn>0</mn><mn>2</mn></msubsup><msqrt><mn>8</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt><mi>dx</mi><mo>-</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mn>3</mn></mfrac><msubsup><mo>|</mo><mn>0</mn><mn>2</mn></msubsup><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><msubsup><mo>∫</mo><mn>0</mn><mn>2</mn></msubsup><msqrt><mn>8</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt><mi>dx</mi><mo>-</mo><mfrac><mn>8</mn><mn>3</mn></mfrac></math></p>
<p>Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mi>sint</mi><mo> </mo><mo>⇒</mo><mo> </mo><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mi>costdt</mi></math></p>
<p><img src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/bai-3-EzoJBv.png" alt="" width="523" height="492" /></p>
<p>Đổi cận:</p>
<table style="border-collapse: collapse; width: 18.4073%; height: 60.7916px;" border="1">
<tbody>
<tr style="height: 22.3958px;">
<td style="width: 22.0674%; height: 22.3958px;">x</td>
<td style="width: 77.3979%; height: 22.3958px;">0 2</td>
</tr>
<tr style="height: 38.3958px;">
<td style="width: 22.0674%; height: 38.3958px;">t</td>
<td style="width: 77.3979%; height: 38.3958px;">0 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math></td>
</tr>
</tbody>
</table>
<p>Suy ra: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>2</mn></msubsup><msqrt><mn>8</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></msubsup><msqrt><mn>8</mn><msup><mi>cos</mi><mn>2</mn></msup><mi mathvariant="normal">t</mi></msqrt><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mi>costdt</mi><mo> </mo><mo>=</mo><mo> </mo><mn>8</mn><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></msubsup><msup><mi>cos</mi><mn>2</mn></msup><mi>tdt</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></msubsup><mo>(</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mn>2</mn><mi mathvariant="normal">t</mi><mo>)</mo><mi>dt</mi><mo>=</mo><mn>4</mn><mo>(</mo><mi mathvariant="normal">t</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>sin</mi><mn>2</mn><mi mathvariant="normal">t</mi><mo>)</mo><mo> </mo><msubsup><mo>|</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></msubsup><mo>=</mo><mn>4</mn><mo>(</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>2</mn></math></p>
<p>Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">S</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>4</mn><mo>-</mo><mfrac><mn>8</mn><mn>3</mn></mfrac><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>+</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>4</mn></mrow><mn>3</mn></mfrac></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>2</mn></msub><mo>=</mo><mn>8</mn><mi mathvariant="normal">π</mi><mo>-</mo><msub><mi mathvariant="normal">S</mi><mn>1</mn></msub><mo>=</mo><mn>6</mn><mi mathvariant="normal">π</mi><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>18</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>4</mn></mrow><mn>3</mn></mfrac></math></p>
<p>Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi mathvariant="normal">S</mi><mn>2</mn></msub><msub><mi mathvariant="normal">S</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><mrow><mn>18</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>4</mn></mrow><mrow><mn>6</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>2</mn></mrow><mrow><mn>3</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math>.</p>
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