Problem
Compute or approximate the corresponding function values and derivative values for the given area function. \[ F(x)=\int_{0}^{\pi} \sqrt{r^{2}+2 r} d r \] In some coses, approximations will neod to be done by using a Riemann sum. For approximations for the given estimates, use the right-endpoint approximation $\boldsymbol{R}_{25}$ for $\mathrm{N}=25$. Determine if the funclion values and derivative values are correct. If not, identify the correct values using the answer bank. \[ \begin{array}{l} F(0)=0 \\ F(4)=\sqrt{24} \\ F^{\prime}(0)=\text { DNE } \\ F^{\prime}(4) \text { \& } 1,021 \end{array} \]
Solution
Step 1 ：Define the integrand function \(f(r) = \sqrt{r^2 + 2r}\).
Step 2 ：Compute the function values \(F(0)\) and \(F(4)\) by integrating the function \(f(r)\) from 0 to \(\pi\).
Step 3 ：Compute the derivative values \(F'(0)\) and \(F'(4)\) by evaluating the function \(f(r)\) at 0 and 4 respectively.
Step 4 ：The computed function values are \(F(0) = 7.27\) and \(F(4) = 7.27\).
Step 5 ：The computed derivative values are \(F'(0) = 0\) and \(F'(4) = 4.90\).
Step 6 ：The given function values and derivative values are incorrect. The correct values are \(F(0) = \boxed{7.27}\), \(F(4) = \boxed{7.27}\), \(F'(0) = \boxed{0}\), and \(F'(4) = \boxed{4.90}\).
