just want to ask what will i do first if i will be given a problem like this:
$\displaystyle \int_7^7 \sqrt[3](|t|+1)dt$
btw the lower limit must be -7 and the |t|+1 must be inside the cube root. I dont know how to put in in latex sorry
thanks
just want to ask what will i do first if i will be given a problem like this:
$\displaystyle \int_7^7 \sqrt[3](|t|+1)dt$
btw the lower limit must be -7 and the |t|+1 must be inside the cube root. I dont know how to put in in latex sorry
thanks
You will need to treat $\displaystyle |t|$ as a hybrid function.
Note that
$\displaystyle |t| = \begin{cases}\phantom{-}t\textrm{ if }t \geq 0\\ -t\textrm{ if }t < 0\end{cases}$.
Therefore
$\displaystyle \int_{-7}^7{\sqrt[3]{|t| + 1}\,dt} = \int_{-7}^0{\sqrt[3]{-t + 1}\,dt} + \int_0^7{\sqrt[3]{t + 1}\,dt}$.
Can you go from here? You will need to use $\displaystyle u$ substitutions...
$\displaystyle \displaystyle \int_{-7}^7 \sqrt[3]{|t|+1}dt$
note that for $\displaystyle f(t) = \sqrt[3]{|t|+1}$ , $\displaystyle f(t) = f(-t)$ , an even function.
using symmetry about the y-axis ...
$\displaystyle \displaystyle \int_{-7}^7 \sqrt[3]{|t|+1}\,dt = 2 \int_0^7 \sqrt[3]{t+1}\,dt = \frac{3}{2}\left[(t+1)^{\frac{4}{3}\right]_0^7 = \frac{45}{2}$