1. **State the problem:** Find the volume of the solid formed by rotating the region bounded by the curves $x = -8 + y^2$ and $x = -2y$ about the line $x = -9$. 2. **Identify the method:** We use the washer method for rotation about a vertical line. The volume is given by $$V = \pi \int_{y=a}^{y=b} \left(R(y)^2 - r(y)^2\right) dy$$ where $R(y)$ is the outer radius and $r(y)$ is the inner radius from the axis of rotation. 3. **Find the intersection points:** Set $-8 + y^2 = -2y$ to find limits: $$y^2 + 2y - 8 = 0$$ Factor: $$ (y+4)(y-2) = 0 \implies y = -4, 2$$ 4. **Determine radii:** The axis of rotation is $x = -9$. - Outer radius $R(y)$ is the distance from $x = -9$ to the farther curve from the axis. - Inner radius $r(y)$ is the distance from $x = -9$ to the nearer curve. For each $y$, the curves give $x$ values: - $x_1 = -8 + y^2$ - $x_2 = -2y$ Since $x_1$ and $x_2$ vary, check which is farther from $x=-9$: Distance from axis: $$d_1 = |x_1 - (-9)| = |y^2 - (-8 + 9)| = |y^2 + 1| = y^2 + 1$$ $$d_2 = |x_2 - (-9)| = |-2y + 9|$$ Check at $y=0$: $d_1=1$, $d_2=9$ so $d_2 > d_1$. At $y=2$: $d_1=2^2+1=5$, $d_2=|-4+9|=5$ equal. At $y=-4$: $d_1=16+1=17$, $d_2=|8+9|=17$ equal. Between $-4$ and $2$, $d_2$ is outer radius and $d_1$ is inner radius. 5. **Set up the integral:** $$V = \pi \int_{-4}^{2} \left((-2y + 9)^2 - (y^2 + 1)^2\right) dy$$ 6. **Expand the squares:** $$(-2y + 9)^2 = 4y^2 - 36y + 81$$ $$(y^2 + 1)^2 = y^4 + 2y^2 + 1$$ 7. **Write the integrand:** $$4y^2 - 36y + 81 - (y^4 + 2y^2 + 1) = -y^4 + 2y^2 - 36y + 80$$ 8. **Integrate term-by-term:** $$\int_{-4}^{2} (-y^4 + 2y^2 - 36y + 80) dy = \left[-\frac{y^5}{5} + \frac{2y^3}{3} - 18y^2 + 80y \right]_{-4}^{2}$$ 9. **Evaluate at $y=2$:** $$-\frac{2^5}{5} + \frac{2 \cdot 2^3}{3} - 18 \cdot 2^2 + 80 \cdot 2 = -\frac{32}{5} + \frac{16}{3} - 72 + 160$$ 10. **Evaluate at $y=-4$:** $$-\frac{(-4)^5}{5} + \frac{2(-4)^3}{3} - 18(-4)^2 + 80(-4) = -\frac{-1024}{5} + \frac{2(-64)}{3} - 18(16) - 320 = \frac{1024}{5} - \frac{128}{3} - 288 - 320$$ 11. **Calculate the difference:** $$\left(-\frac{32}{5} + \frac{16}{3} - 72 + 160\right) - \left(\frac{1024}{5} - \frac{128}{3} - 288 - 320\right)$$ Simplify stepwise: $$= \left(-\frac{32}{5} + \frac{16}{3} + 88\right) - \left(\frac{1024}{5} - \frac{128}{3} - 608\right)$$ $$= -\frac{32}{5} + \frac{16}{3} + 88 - \frac{1024}{5} + \frac{128}{3} + 608$$ $$= \left(-\frac{32}{5} - \frac{1024}{5}\right) + \left(\frac{16}{3} + \frac{128}{3}\right) + (88 + 608)$$ $$= -\frac{1056}{5} + \frac{144}{3} + 696 = -211.2 + 48 + 696 = 532.8$$ 12. **Multiply by $\pi$ to get volume:** $$V = \pi \times 532.8 \approx 1673.362$$ **Final answer:** The volume of the solid is approximately **1673.362** (rounded to the nearest thousandth).