Area between lines

From the graph below, we can see that we just need to find the area between $y = 2 \sqrt{x}$ and $y = 3$ from $x = 5$ to $x = 10$ , and then subtract the area of the cross-hatched triangle, which is clearly $\frac{1}{2}(2)(2) = 2$ .

So the area we're looking for is

$\begin{array}{rl} \int_5^{10} (2\sqrt{x} - 3) \ dx - 2 &= \ \ \left[ \dfrac{4}{3} \sqrt{x^3} - 3x \right]_5^{10} - \ 2 \\ \\ &= \ \ \left( \dfrac{40 \sqrt{10}}{3} - 30 \right) - \left( \dfrac{20 \sqrt{5}}{3} - 15 \right) - \ 2 \\ \\ &= \ \ \dfrac{40 \sqrt{10} - 20 \sqrt{5} }{3} - 17 \\ \\ &\approx \ \ \boxed{10.26} \end{array}$