Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves \( x=0, y=1, x=y^{6} \), about the line \( y=1 \). l

To find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves \( x=0 \), \( y=1 \), and \( x=y^{6} \) about the line \( y=1 \), we can use the method of cylindrical shells. ### Step 1: Identify the region of integration The curves are: - \( x = 0 \) (the y-axis) - \( y = 1 \) (a horizontal line) - \( x = y^6 \) (a curve) To find the intersection points, we set \( y = 1 \) in \( x = y^6 \): \[ x = 1^6 = 1 \] Thus, the region of integration is bounded by \( y = 0 \) to \( y = 1 \) and \( x = y^6 \). ### Step 2: Set up the volume integral Using the method of cylindrical shells, the volume \( V \) is given by the formula: \[ V = 2\pi \int_{a}^{b} (radius)(height) \, dy \] In this case: - The radius is the distance from the line \( y = 1 \) to the curve, which is \( 1 - y \). - The height is given by the function \( x = y^6 \). Thus, the volume integral becomes: \[ V = 2\pi \int_{0}^{1} (1 - y)(y^6) \, dy \] ### Step 3: Simplify the integral Now we simplify the integrand: \[ V = 2\pi \int_{0}^{1} (y^6 - y^7) \, dy \] ### Step 4: Calculate the integral Now we compute the integral: \[ V = 2\pi \left[ \frac{y^7}{7} - \frac{y^8}{8} \right]_{0}^{1} \] Evaluating at the bounds: \[ V = 2\pi \left( \frac{1^7}{7} - \frac{1^8}{8} \right) - 2\pi \left( \frac{0^7}{7} - \frac{0^8}{8} \right) \] \[ = 2\pi \left( \frac{1}{7} - \frac{1}{8} \right) \] ### Step 5: Find a common denominator The common denominator of 7 and 8 is 56: \[ \frac{1}{7} = \frac{8}{56}, \quad \frac{1}{8} = \frac{7}{56} \] Thus, \[ \frac{1}{7} - \frac{1}{8} = \frac{8}{56} - \frac{7}{56} = \frac{1}{56} \] ### Step 6: Final volume calculation Substituting back into the volume expression: \[ V = 2\pi \left( \frac{1}{56} \right) = \frac{2\pi}{56} = \frac{\pi}{28} \] ### Final Answer The volume of the solid obtained by rotating the region about the line \( y=1 \) is: \[ \boxed{\frac{\pi}{28}} \]