What is the integral from 1 to 4 of |-sqrt(x)-(2-x)| ?

The integral from 1 to 4 of |-sqrt(x)-(2-x)| is 19/6

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integral from 1 to 4 of |-sqrt(x)-(2-x)|

\int_{1}^{4} - x - (2 - x) d x

\frac{19}{6}

Decimal Notation

3.16666 \dots

Solution steps

\int_{1}^{4} - x - (2 - x) d x

Eliminate Absolutes

= \int_{1}^{4} x + 2 - x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{4} x d x + \int_{1}^{4} 2 d x - \int_{1}^{4} x d x

\int_{1}^{4} x d x = \frac{14}{3}

\int_{1}^{4} 2 d x = 6

\int_{1}^{4} x d x = \frac{15}{2}

= \frac{14}{3} + 6 - \frac{15}{2}

Simplify

\frac{14}{3} + 6 - \frac{15}{2} : \frac{19}{6}

= \frac{19}{6}

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\int_{- 2}^{2} x^{4} (e^{x} - e^{- x}) d x

\int_{2 x + 3}^{x^{3}} e^{- t^{2}} d t

What is the integral from 1 to 4 of |-sqrt(x)-(2-x)| ?
The integral from 1 to 4 of |-sqrt(x)-(2-x)| is 19/6