Problem
1. [2/3 Points] DETAILS PREVIOUS ANSWERS SCALC9 5.2.011. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line. $y=x+1, y=0, x=0, x=5 ;$ about the $x$-axis \[ v= \]
Solution
Step 1 ：The volume of the solid obtained by rotating a region bounded by the curves \(y=f(x)\) and \(y=g(x)\), where \(f(x) \geq g(x)\), about the x-axis from \(x=a\) to \(x=b\) is given by the formula: \[V = \pi \int_{a}^{b} [f(x)]^2 - [g(x)]^2 dx\]
Step 2 ：In this case, the region is bounded by the curves \(y=x+1\) and \(y=0\) from \(x=0\) to \(x=5\). So, we have \(f(x) = x+1\) and \(g(x) = 0\).
Step 3 ：We can substitute these into the formula and evaluate the integral to find the volume.
Step 4 ：Final Answer: The volume of the solid obtained by rotating the region bounded by the curves \(y=x+1\), \(y=0\), \(x=0\), and \(x=5\) about the x-axis is \(\boxed{\frac{215\pi}{3}}\).
