If f(x)= { ((1−x, 0≤x≤1)),((0, 1≤x≤2 )),(((2−x)^2 , 2≤x≤3)) :} and φ(x)=∫_( 0) ^x f(t) dt. Then for any x ∈ [2, 3], φ(x) =


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Question Number 88210 by bagjamath last updated on 09/Apr/20
$If f(x)= { ((1−x, 0≤x≤1)),((0, 1≤x≤2 )),(((2−x)^2 , 2≤x≤3)) :} and φ(x)=∫_( 0) ^x f(t) dt. Then for any x ∈ [2, 3], φ(x) =$
$If f (x) = {\begin{cases} 1 - x, 0 ⩽ x ⩽ 1 \\ 0, 1 ⩽ x ⩽ 2 \\ {(2 - x)}^{2}, 2 ⩽ x ⩽ 3 \end{cases} and$ $ϕ (x) = \underset{0}{\int^{x}} f (t) d t . Then for any x \in [2, 3],$ $ϕ (x) =$
	Answered by MJS last updated on 09/Apr/20

	$= \underset{0}{\int^{1}} (1 - t) d t + 0 \underset{1}{\int^{2}} d t + \underset{2}{\int^{x}} {(2 - t)}^{2} d t =$ $= {[t - \frac{1}{2} t^{2}]}_{0}^{1} + {[4 t - 2 t^{2} + \frac{t^{3}}{3}]}_{2}^{x} =$ $= \frac{1}{3} x^{3} - 2 x^{2} + 4 x - \frac{13}{6}$
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