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Giải bài tập 22 trang 92 SGK Toán 12 tập 2 - Kết nối tri thức

Gọi (H) là hình phẳng giới hạn bởi các đường \(y = - {x^2} + 2x;y = 0,x = 0\) và \(x = 2\). Tính thể tích của khối tròn xoay thu được khi quay hình phẳng (H) xung quanh trục Ox.

Tổ🅠ng 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

 
 
Gọi (H) là hình phẳng giới hạn bởi các đường \(y =  - {x^2} + 2x;y = 0,x = 0\) và \(x = 2\). Tính thể tích của khối tròn xoay thu được khi quay hình phẳng (H) xung quanh trục Ox.
 

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về công thức tính thể tích của khối tròn xoay để tính: Cho hàm số f(x) liên tục, không âm trên đoạn [a; b]. Khi quay hình phẳng giới hạn bởi đồ thị hàm số \(y = f\left( x \right)\), trục hoành và hai đường thẳng \(x = a,x = b\) xung quanh trục hoành, ta được hình khối gọi là một khối tròn xoay. Khi cắt khối tròn xoay đó bởi một mặt phẳng vuông góc với trục Ox tại điểm \(x \in \left[ {a;b} \right]\) được một hình tròn có bán kính f(x). Thể tích của khối tròn xoay này là: \(V = \pi \int\limits_a^b {{f^2}\left( x \right)dx} \). 
 

Lời giải chi tiết

Thể tích khối tròn xoay cần tính là: \(V = \pi \int\limits_0^2 {{{\left( { - {x^2} + 2x} \right)}^2}dx}  = \pi \int\limits_0^2 {\left( {{x^4} - 4{x^3} + 4{x^2}} \right)dx} \) \( = \pi \left( {\frac{{{x^5}}}{5} - {x^4} + \frac{{4{x^3}}}{3}} \right)\left| \begin{array}{l}2\\0\end{array} \right. = \pi \left( {\frac{{32}}{5} - 16 + \frac{{32}}{3}} \right) = \frac{{16\pi }}{{15}}\)
 

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