To find the volume of the bowl formed by revolving the area enclosed by the curves \( y = x^2 - 1 \), \( y = 3 \), and \( x \geq 0 \) around the y-axis, we will use the method of cylindrical shells. Let's break down the steps to arrive at the correct answer. ### Step 1: Identify the curves and the region of interest 1. **Curves**: We have two curves: - \( y = x^2 - 1 \) (a parabola opening upwards) - \( y = 3 \) (a horizontal line) 2. **Intersection Points**: To find the region enclosed by these curves, we need to find their points of intersection. Set \( y = x^2 - 1 \) equal to \( y = 3 \): \[ x^2 - 1 = 3 \] \[ x^2 = 4 \quad \Rightarrow \quad x = 2 \quad \text{(since } x \geq 0\text{)} \] The intersection points are at \( (2, 3) \). 3. **Region**: The area we are interested in is bounded by \( y = x^2 - 1 \) from below and \( y = 3 \) from above, for \( x \) values ranging from \( 0 \) to \( 2 \). ### Step 2: Set up the volume integral using the shell method The volume \( V \) of the solid formed by revolving the area around the y-axis can be calculated using the formula for cylindrical shells: \[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \] where \( f(x) \) is the height of the shell, and \( a \) and \( b \) are the bounds of integration. In our case: - The height of the shell is given by the difference between the upper curve and the lower curve: \[ f(x) = 3 - (x^2 - 1) = 4 - x^2 \] - The bounds of integration are \( a = 0 \) and \( b = 2 \). ### Step 3: Calculate the volume Now we can set up the integral: \[ V = 2\pi \int_{0}^{2} x(4 - x^2) \, dx \] ### Step 4: Evaluate the integral 1. **Distribute**: \[ V = 2\pi \int_{0}^{2} (4x - x^3) \, dx \] 2. **Integrate**: \[ V = 2\pi \left[ \frac{4x^2}{2} - \frac{x^4}{4} \right]_{0}^{2} \] \[ = 2\pi \left[ 2x^2 - \frac{x^4}{4} \right]_{0}^{2} \] \[ = 2\pi \left[ 2(2^2) - \frac{(2^4)}{4} \right] \] \[ = 2\pi \left[ 2(4) - \frac{16}{4} \right] \] \[ = 2\pi \left[ 8 - 4 \right] \] \[ = 2\pi \cdot 4 = 8\pi \] ### Conclusion The volume of the bowl formed by revolving the area around the y-axis is \( 8\pi \) cubic units. Therefore, the correct option is: **C. 8π cubic units** ### Explanation of Other Options - **A. 7π cubic units**: This value is too low and does not account for the full height of the region being revolved. - **B. 15π/2 cubic units**: This value is incorrect as it does not match the calculated volume. - **D. 17π/2 cubic units**: This value is also too high and does not correspond to the volume derived from the integral. ### Revision Summary - The volume of a solid of revolution can be calculated using the method of cylindrical shells. - Identify the curves and their intersection points to determine the region of integration. - Set up the integral correctly, considering the height of the shells. - Evaluate the integral step-by-step to find the volume accurately. This thorough approach ensures a clear understanding of how to solve similar problems in the future.