If-f-x-1-x-0-x-1-0-1-x-2-2-x-2-2-x-3-and-x-0-x-f-t-dt-Then-for-any-x-2-3-x-

Question Number 88210 by bagjamath last updated on 09/Apr/20

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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