The diagram shows the curve $y=(x-3{)}^2$ and the line $y=2x-3$ that intersect at points $A$ and $B$. Find the area of the shaded region.     [Enter the value excluding units]

The diagram shows the curve $y=-x^2+11x-18$ and the line $2x+y=12$. Find the area of the shaded region.       [Enter the value excluding units]

Sketch the following curve and line and find the area enclosed between their graph.

$y=x^2-3$ and $y=6$

Sketch the following curve and line and find the area enclosed between their graph.

$y=-x^2+12x-20$ and $y=2x+1$

Sketch the following curve and line and find the area enclosed between their graph.

$y=x^2-4x+4$ and $2x+y=12$

Sketch the curves $y=x^2$ and $y=x(2-x)$ and find the area enclosed between the two curves.

The diagram shows the curve $y=\sqrt{x+4}$ and the line $y=\frac{1}{2}x+2$ meeting at the points $(-4,0)$ and $(0,2)$. Find the area of the shaded region.         [Enter the value excluding units]

Show that the following improper integral has a finite value and find the value.

${\int }_1^{\infty }\frac{2}{x^2} \mathrm{d}x$

Show that the following improper integral has a finite value and find the value.

${\int }_4^{\infty }\frac{4}{x^5} \mathrm{d}x$