Bài 3: Ứng dụng của tích phân trong hình học
Hướng dẫn giải Bài 1 (Trang 121 SGK Toán Giải tích 12)
<p><strong>Bài 1 (Trang 121 SGK Toán Giải tích 12):</strong></p>
<p>Tính diện tích hình phẳng giới hạn bởi các đường:</p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn></math>;</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced open="|" close="|"><mrow><mi>ln</mi><mi>x</mi></mrow></mfenced><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn></math>;</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>6</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></math>.</p>
<p><strong><em>Hướng dẫn giải:</em></strong></p>
<p>a) Phương trình hoành độ giao điểm của hai đường cong là:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo> </mo><mo>⇔</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><mo> </mo><msubsup><mo>[</mo><mrow><mi>x</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></mrow></msubsup></math></p>
<p>Diện tích hình phẳng đã cho là</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mfenced open="|" close="|"><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mi>dx</mi><mo>=</mo><mfenced open="|" close="|"><mrow><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>(</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mo>)</mo><mi>dx</mi></mrow></mfenced><mo>=</mo><msubsup><mfenced open="|" close="|"><mrow><mo>(</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>-</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>)</mo></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></math> (đvdt)</p>
<p>b) Phương trình hoành độ giao điểm của hai đường cong là</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mi>ln</mi><mo> </mo><mi mathvariant="normal">x</mi></mrow></mfenced><mo>=</mo><mn>1</mn><mo>⇔</mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi>ln</mi><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>ln</mi><mo> </mo><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable><mo> </mo><mo>⇔</mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mi mathvariant="normal">e</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">x</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mi mathvariant="normal">e</mi></mfrac></mtd></mtr></mtable></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo>=</mo><msubsup><mo>∫</mo><mfrac><mn>1</mn><mi mathvariant="normal">e</mi></mfrac><mi mathvariant="normal">e</mi></msubsup><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mfenced open="|" close="|"><mi>lnx</mi></mfenced></mrow></mfenced><mi>dx</mi><mo>=</mo><msubsup><mo>∫</mo><mfrac><mn>1</mn><mi mathvariant="normal">e</mi></mfrac><mn>1</mn></msubsup><mo>(</mo><mn>1</mn><mo>+</mo><mi>lnx</mi><mo>)</mo><mi>dx</mi><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mi mathvariant="normal">e</mi></msubsup><mo>(</mo><mn>1</mn><mo>-</mo><mi>lnx</mi><mo>)</mo><mi>dx</mi></math></p>
<p>Tính <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>lnxdx</mi></math></p>
<p>Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi>u</mi><mo>=</mo><mi>ln</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi>d</mi><mi>v</mi><mo>=</mo><mi>d</mi><mi>x</mi></mtd></mtr></mtable><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>d</mi><mi>u</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mi>x</mi></mfrac></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mi>x</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math></p>
<p>Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>ln</mi><mi>x</mi><mi>d</mi><mi>x</mi><mo>=</mo><mi>x</mi><mi>ln</mi><mi>x</mi><mo>-</mo><mo>∫</mo><mi>d</mi><mi>x</mi><mo> </mo><mo>=</mo><mi>x</mi><mi>ln</mi><mi>x</mi><mo>-</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></p>
<p>Vậy một nguyên hàm của y = lnx là <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mi>ln</mi><mi>x</mi><mo>-</mo><mi>x</mi></math></p>
<p>Do đó <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>x</mi><mi>ln</mi><mi>x</mi><msubsup><mo>|</mo><mfrac><mn>1</mn><mi>e</mi></mfrac><mn>1</mn></msubsup><mo> </mo><mo>+</mo><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>x</mi><mi>ln</mi><mi>x</mi><mo>)</mo><msubsup><mo>|</mo><mn>1</mn><mi>e</mi></msubsup><mo>=</mo><mfrac><mn>1</mn><mi>e</mi></mfrac><mo>+</mo><mn>2</mn><mi>e</mi><mo>-</mo><mi>e</mi><mo>-</mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mi>e</mi></mfrac><mo>+</mo><mi>e</mi><mo>-</mo><mn>2</mn></math> (đvdt)</p>
<p>c) Phương trình hoành độ giao điểm của hai đường cong là:</p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>6</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⇔</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>18</mn><mo>=</mo><mn>0</mn><mo>⇔</mo><msubsup><mo>[</mo><mrow><mi>x</mi><mo>=</mo><mn>6</mn></mrow><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow></msubsup></math></p>
<p>Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo>=</mo><msubsup><mo>∫</mo><mn>3</mn><mn>6</mn></msubsup><mfenced open="|" close="|"><mrow><msup><mrow><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>6</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mo>(</mo><mn>6</mn><mi mathvariant="normal">x</mi><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>)</mo></mrow></mfenced><mi>dx</mi><mo>=</mo><mfenced open="|" close="|"><mrow><msubsup><mo>∫</mo><mn>3</mn><mn>6</mn></msubsup><mn>2</mn><mo>(</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>18</mn><mo>)</mo><mi>dx</mi></mrow></mfenced><mo>=</mo><mn>9</mn></math> (đvdt).</p>
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