Find the shaded area enclosed by the curve $y=2\sqrt{x}$, the line $x+y=8$ and the $x-$axis.       [Enter the value excluding units]

The diagram shows the curve $y=\sqrt{2x+1}$ that intersects the $x-$axis at $A$. The normal to the curve at $B(4,3)$ meets the $x-$axis at $C$. Find the area of the shaded region.        [Enter the value excluding units]

The figure shows part of the curve $y=f\left(x\right)$. The points $P(2,4)$ and $Q(7,12)$ lie on the curve. Given that ${\int }_2^{7}y \mathrm{d}x=42$, find the value of ${\int }_4^{12}x \mathrm{d}y$.        [Enter the value excluding units]

The figure shows part of the curve $y=g\left(x\right)$. The points $A(2,8)$ and $B(6,1)$ lie on the curve. Given that ${\int }_2^{6}y \mathrm{d}x=16$, find the value of ${\int }_1^{8}x \mathrm{d}y$.       [Enter the value excluding units]

Find the area of the region bounded by the curve $y=5+6x-x^2$, the line $x=4$ and the line $y=5$.          [Enter the value excluding units]

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 $y=2x+1$