Problem
The regions \( \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D} \), and \( \mathrm{E} \) in the figure below are bounded by the graph of the function \( f \) and the \( x \)-axis. The areas of each enclosed region are labeled in the figure with their respective numerical values. What is the value of \( \int_{-2}^{3}(f(x)+1) d x ? \)
Solution
Step 1 ：\( \int_{-2}^{3}(f(x)+1) d x = \int_{-2}^{3}f(x) d x + \int_{-2}^{3}1 d x \)
Step 2 ：\( \int_{-2}^{3}f(x) d x = (4+8+20)-(-1+2+15) = 9 \)
Step 3 ：\( \int_{-2}^{3}1 d x = 3 - (-2) = 5 \)
Step 4 ：\( \int_{-2}^{3}(f(x)+1) d x = 9 + 5 = 14 \)
