Find the area of the region in the first quadrant bounded by the line $y=2 x$, the line $x=5$, the curve $y=\frac{2}{x^{2}}$, and the $x$-axis.
Solution
Step 1 ：We are given the line \(y=2x\), the line \(x=5\), the curve \(y=\frac{2}{x^{2}}\), and the x-axis. We are asked to find the area of the region in the first quadrant bounded by these lines and curve.
Step 2 ：The area of the region can be found by integrating the difference between the two functions from the point where they intersect to the line \(x=5\).
Step 3 ：The two functions intersect when \(2x = \frac{2}{x^{2}}\), which simplifies to \(x^{3} = 1\), so \(x = 1\).
Step 4 ：Therefore, the area of the region is given by the integral from 1 to 5 of \(2x - \frac{2}{x^{2}}\).
Step 5 ：By calculating the integral, we find that the area of the region is \(\frac{112}{5}\).
Step 6 ：Final Answer: The area of the region in the first quadrant bounded by the line \(y=2 x\), the line \(x=5\), the curve \(y=\frac{2}{x^{2}}\), and the x-axis is \(\boxed{\frac{112}{5}}\).
