(6) \( \lim _{t \rightarrow 0}\left(\frac{1}{t}-\frac{1}{t^{2}+t}\right) \), (7) Let \( f(x)=\left\{\begin{array}{ll}x^{2}+1 & \text { if } x<1 \\ (x-2)^{2} & \text { if } x \geq 1\end{array}\right. \) (a) Find \( \lim _{x \rightarrow 1^{-}} f(x) \) and \( \lim _{x \rightarrow 1^{+}} f(x) \), (b) Does \( \lim _{x \rightarrow 1} f(x) \) exist? (8) Let \( g(x)=\frac{x^{2}+x-6}{|x-2|} \). (a) Find \( \lim _{x \rightarrow 2^{+}} g(x) \) and \( \lim _{x \rightarrow 2^{-}} g(x) \). (b) Does \( \lim _{x \rightarrow 2} g(x) \) exist?

The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = \(x\) and height = \(3 - x\) for \(0 \leq x \leq 3\). Calculate the volume of the solid when these rectangles are stacked from \(x=0\) to \(x=3\).

\( \int _ { 0 } ^ { 2 } d y \int _ { y - 2 } ^ { 2 - y } ( 2 x - y ) d y \)

Find the volume of the solid formed by rotating the area under the curve \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 3 \) about the y-axis.

Find the are length of the curve \( y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}} \) from \( x=1 \) to \( x=2 \)

If a car has an acceleration function of \( a(t) = 6t \), what is its velocity function assuming it starts from rest?