Let's go step by step through the problem: ### **Given:** - Curves: \( y = x - 1 \), \( y = 4 - x \), and \( x = 0 \) - Solid obtained by rotating the enclosed region about the **y-axis** - **Method:** Cylindrical Shells --- ## **(a) Limits of Integration** ### **Step 1: Find the intersection points** Set \( y = x - 1 \) equal to \( y = 4 - x \): \[ x - 1 = 4 - x \] \[ 2x = 5 \] \[ x = \frac{5}{2} \] Check where \( x = 0 \) fits in: - At \( x = 0 \), \( y = -1 \) from \( y = x - 1 \) - At \( x = \frac{5}{2} \), both curves meet at \( y = \frac{3}{2} \) So, the **limits for \( x \)** are from \( 0 \) to \( \frac{5}{2} \). **Final Answer for (a):** \[ \boxed{0 \leq x \leq \frac{5}{2}} \] --- ## **(b) Cylindrical Shells' Base Radius as a Function of \( x \)** The **cylindrical shell** at position \( x \) has radius = distance from the y-axis (since we rotate around y-axis): \[ \text{Radius} = x \] **Final Answer for (b):** \[ \boxed{\text{Base radius as a function of } x: \quad r(x) = x} \] --- ## **(c) Cylindrical Shells' Height as a Function of \( x \)** For each \( x \), the height is the vertical distance between the two curves: \[ \text{Height} = (\text{Upper curve}) - (\text{Lower curve}) \] - Upper curve: \( y = 4 - x \) - Lower curve: \( y = x - 1 \) So: \[ \text{Height} = (4 - x) - (x - 1) = 4 - x - x + 1 = 5 - 2x \] **Final Answer for (c):** \[ \boxed{\text{Height as a function of } x: \quad h(x) = 5 - 2x} \] --- ## **Summary Table** | Part | Answer | |------|--------| | (a) | \( 0 \leq x \leq \frac{5}{2} \) | | (b) | \( r(x) = x \) | | (c) | \( h(x) = 5 - 2x \) | Let me know if you'd like to compute the actual volume as well!