What is the value of \(\rm \displaystyle \int_{-2}^2 |x|dx\) ?

Question

Question

Answer 1

Concept:

Definite Integral: If ∫ f(x)dx = g(x) + C, then \(\rm \displaystyle \int_a^b f(x)dx = [ g(x)]_a^b=g(b)-g(a)\).
\(\rm \displaystyle \int_a^b f(x)dx = \int_a^c f(x)dx+\int_c^b f(x)dx\) for some number c between a and b.
\(\rm \displaystyle \int x^ndx=\dfrac{x^{n+1}}{n+1}\).
\(\rm |x|=\left\{\begin{matrix}\rm \ \ x, &\rm x>0\\\rm \ \ 0, & \rm x=0 \\\rm -x, &\rm x<0\\\end{matrix}\right.\).

Calculation:

\(\rm \displaystyle \int_{-2}^2 |x|dx = \int_{-2}^0 |x|dx+\int_{0}^2 |x|dx\)

\(\rm \displaystyle =\int_{-2}^0 -xdx+\int_{0}^2 xdx\)

\(\rm =-\dfrac{1}{2}[x^2]_{-2}^{\ \ \ 0}+\dfrac{1}{2}[x^2]_0^2\)

\(\rm =\dfrac{-1}{2}[0^2-(-2)^2]+\dfrac{1}{2}[2^2-0^2]\)

= 2 + 2 = 4