1. **Problem Statement**: We are asked to find the values of A, B, C, D, and E in the integral expression $$V = \int_A^B 2\pi (1 + C)(D - E) \, dx$$ which represents the volume of the solid generated by rotating the region bounded by y = x^2 + 2x + 1, x = 1, and y = -1 around the line x = 1. 2. **Finding limits A and B**: The region R is bounded vertically by the curve and y = -1. Since the given curve is $$y = x^2 + 2x + 1 = (x+1)^2,$$ and y = -1 equal when $$(x+1)^2 = -1$$ normally no real solution, but the bounding x values come from the vertical boundary $x = 1$ The region extends from the left to the line $x=1$. We find the intersection of the curve with $y=-1$ for x-values: $$(x+1)^2 = -1$$ no real solution. The lowest y-bound is at $y=-1$, so x limits are determined by the curve and $x=1$. Given the curve vertex is at $x=-1$, and axis of symmetry is $x=-1$, region runs from $x = -1$ (where $y=0$) to $x=1$. So: $$A = -1$$ $$B = 1$$ 3. **Finding C**: We are revolving around the vertical line $x=1$. The radius of each washer/disk is the horizontal distance from $x=1$ to the curve. Radius $$r = 1 - x$$ As such, $$(1 + C) = 1 - x \Rightarrow C = -x$$ 4. **Determining D and E**: Since we revolve the vertical slice at $x$ with height from $y=-1$ (lower curve) to $y = (x+1)^2$ (upper curve) The thickness is $dx$, and the height difference in $y$ is: $$D - E = y_{upper} - y_{lower} = (x^2 + 2x +1) - (-1) = x^2 + 2x +2$$ Thus: $$D = x^2 + 2x +1$$ $$E = -1$$ 5. **Final integral representation**: $$V = \int_{-1}^1 2\pi (1 - x)\big((x^2 + 2x + 1) - (-1)\big) dx = \int_{-1}^1 2\pi (1 - x)(x^2 + 2x + 2) dx$$ **Summary of choices:** 1. A = -1 2. B = 1 3. C = -x 4. D = x^2 + 2x + 1 5. E = -1