Final answer:
To find the volume of the solid formed by rotating the region enclosed by the lines x=0, x=1, y=0, and y=7x⁹, we can use the disk method. We need to find the limits of integration, which are from y=0 to y=7x⁹. Next, we need to find the equation of the curve y=7x⁹ in terms of x.
Step-by-step explanation:
To find the volume of the solid formed by rotating the region enclosed by the lines x=0, x=1, y=0, and y=7x⁹, we can use the disk method. First, we need to find the limits of integration, which are from y=0 to y=7x⁹. Next, we need to find the equation of the curve y=7x⁹ in terms of x.
We can rewrite this equation as x=(y/7)^(1/9). To find the volume, we integrate the area of each disk from y=0 to y=7x⁹ and sum the results. The formula for the volume using the disk method is V=π∫(R(y))² dy, where R(y) is the radius of each disk.
Plugging in the equation for x, we have R(y)=(y/7)^(1/9). Therefore, the formula for the volume becomes V=π∫((y/7)^(1/9))² dy. Evaluating the integral from y=0 to y=7x⁹, we get V=π∫(y^(2/9))/49 dy.
After integrating and simplifying, we find V=(49π/9)7^(11/9). Evaluating this expression, we get V≈16.44 units cubed.