H.W)Let \( f(x, y)=x y, \quad C: \vec{r}_{C}(t)=(t) i+(2 t) j, 0 \leq t \leq 1 \), evalaute \( I_{L, x C} \) and \( I_{L, x, C} \) Answer \( =\frac{2}{3},-\frac{2}{3} \)

The integral in this exercise converges. Evaluate the integral without using a table. \( \int_{14}^{\infty} \frac{2}{v^{2}-v} d v \)

Paraffin is stored in a tank with a horizontal base. At time \( t \) minutes, the depth of paraffin in the tank is \( x \mathrm{~cm} \). When \( t=0, x=72 \). There is a tap in the side of the tank through which the paraffin can flow. When the tap is opened, the flow of the paraffin is modelled by the differential equation \[ \frac{\mathrm{d} x}{\mathrm{~d} t}=-4(x-8)^{\frac{1}{3}} \text {. } \] (i) How long does it take for the level of paraffin to fall from a depth of 72 cm to a depth of 35 cm ? (ii) The tank is filled again to its original depth of 72 cm of paraffin and the tap is then opened. The paraffin flows out until it stops. How long does this take?

A container in the shape of an inverted cone of radius 3 metres and vertical height 4.5 metres is initially filled with liquid fertiliser. This fertiliser is released through a hole in the bottom of the container at a rate of \( 0.01 \mathrm{~m}^{3} \) per second. At time \( t \) seconds the fertiliser remaining in the container forms an inverted cone of height \( h \) metres. [The volume of a cone is \( V=\frac{1}{3} \pi r^{2} h \).] (i) Show that \( h^{2} \frac{\mathrm{~d} h}{\mathrm{~d} t}=-\frac{9}{400 \pi} \). (ii) Express \( h \) in terms of \( t \). (iii) Find the time it takes to empty the container, giving your answer to the nearest minute.

\( \lim \frac { 3 n ^ { 3 } - 2 n + 1 } { 4 n ^ { 4 } + 2 n + 1 } \)

(f) \( \lim _{x \mapsto 0^{-}} \frac{-2 \cdot x^{2}}{x^{4}+x^{3}} \)