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Giải bài tập 21 trang 92 SGK Toán 12 tập 2 - Kết nối tri thức

Tính diện tích hình phẳng giới hạn bởi các đường \(y = {x^2} - 1;y = x + 5,x = - 2,x = 3\).

🍌Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

 
 
Tính diện tích hình phẳng giới hạn bởi các đường \(y = {x^2} - 1;y = x + 5,x =  - 2,x = 3\).
 

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về diện tích hình phẳng giới hạn bởi hai đồ thị hàm số và đường thẳng \(x = a,x = b\) để tính: Diện tích S của hình phẳng giới hạn đồ thị của hai hàm số f(x), g(x) liên tục trên đoạn [a; b] và hai đường thẳng \(x = a,x = b\), được tính bằng công thức \(S = \int\limits_a^b {\left| {f\left( x \right) - g\left( x \right)} \right|dx} \).

 

Lời giải chi tiết

Diện tích hình cần tính là: \(S = \int\limits_{ - 2}^3 {\left| {\left( {x + 5} \right) - \left( {{x^2} - 1} \right)} \right|dx}  = \int\limits_{ - 2}^3 {\left( { - {x^2} + x + 6} \right)dx} \) \( = \left( {\frac{{ - {x^3}}}{3} + \frac{{{x^2}}}{2} + 6x} \right)\left| \begin{array}{l}3\\ - 2\end{array} \right. = \frac{{27}}{2} + \frac{{22}}{3} = \frac{{125}}{6}\)
 

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