a. Pt hoành độ giao điểm: \(x^3-4x+3=0\Rightarrow\left[{}\begin{matrix}x=1\\x=3\end{matrix}\right.\)
\(S=\int\limits^2_0\left|x^2-4x+3\right|dx=\int\limits^1_0\left(x^2-4x+3\right)dx-\int\limits^2_1\left(x^2-4x+3\right)dx\)
\(=\left(\dfrac{1}{3}x^3-2x^2+3x\right)|^1_0-\left(\dfrac{1}{3}x^3-2x^2+3x\right)|^2_1=2\)
b. Pt hoành độ giao điểm:
\(cosx=0\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}\\x=\dfrac{3\pi}{2}\end{matrix}\right.\)
\(S=\int\limits^{2\pi}_0\left|cosx\right|dx=\int\limits^{\dfrac{\pi}{2}}_0cosxdx-\int\limits^{\dfrac{3\pi}{2}}_{\dfrac{\pi}{2}}cosxdx+\int\limits^{2\pi}_{\dfrac{3\pi}{2}}cosxdx\)
\(=1-\left(-2\right)+1=4\)
c.
Ta có \(x^3-3x+6>0\) ; \(\forall x\in\left[1;3\right]\)
\(S=\int\limits^3_1\left|x^3-3x+6\right|dx=\int\limits^3_1\left(x^3-3x+6\right)dx=\left(\dfrac{1}{4}x^4-\dfrac{3}{2}x^2+6x\right)|^3_1=20\)
d.
\(x^3-3x+1>0\) ; \(\forall x\in\left[-1;0\right]\)
\(S=\int\limits^0_{-1}\left|x^3-3x+1\right|dx=\int\limits^0_{-1}\left(x^3-3x+1\right)dx=\left(\dfrac{1}{4}x^4-\dfrac{3}{2}x^2+x\right)|^0_{-1}=\dfrac{9}{4}\)
e.
Pt hoành độ giao điểm: \(x^3-x=0\Rightarrow\left[{}\begin{matrix}x=0\\x=-1\\x=1\end{matrix}\right.\)
\(S=\int\limits^1_{-1}\left|x^3-x\right|dx=\int\limits^0_{-1}\left(x^3-x\right)dx-\int\limits^1_0\left(x^3-x\right)dx=\dfrac{1}{2}\)
f. Pt hoành độ giao điểm:
\(e^x\left(x-1\right)=0\Leftrightarrow x=1\)
\(S=\int\limits^1_0\left|e^x\left(x-1\right)\right|dx=\int\limits^1_0\left(1-x\right)e^xdx=e-2\)
g. Pt hoành độ giao điểm: \(xlnx=0\Leftrightarrow x=1\)
\(S=\int\limits^2_1\left|xlnx\right|dx=\int\limits^2_1xlnxdx=ln4-\dfrac{3}{4}\)
