1. V = ∫[π(x²+5x^6)^2] dy from y = 0 to y = b, where b is the upper bound of the y-axis.
2. V = ∫[π((1 - (y-4)²)²)] dx from x = a to x = 1, where a is the lower bound of the x-axis.
1:
Identify the curves and the axis of rotation. In this case, the curves are y = x²+5x^6 and y = 0, and the axis of rotation is the x-axis.
Imagine the region bounded by the curves being sliced into thin horizontal disks. Each disk, when rotated about the x-axis, forms a cylinder.
Express the volume of each cylinder as a function of its radius and thickness. The radius is equal to the distance between the curve y = x²+5x^6 and the x-axis at a specific y-value. The thickness is equal to the difference between two consecutive y-values.
Sum the volumes of infinitely many such cylinders using a definite integral. This integral represents the volume of the entire solid formed by rotating the region about the x-axis.
2:
Identify the curves and the axis of rotation. In this case, the curves are x = (y-4)² and x = 1, and the axis of rotation is the line y = 3.
Imagine the region bounded by the curves being sliced into thin vertical disks. Each disk, when rotated about the y-axis, forms a cylinder.
Express the volume of each cylinder as a function of its radius and thickness. The radius is equal to the distance between the curve x = 1 and the curve x = (y-4)², at a specific x-value. The thickness is equal to the difference between two consecutive x-values.
Sum the volumes of infinitely many such cylinders using a definite integral. This integral represents the volume of the entire solid formed by rotating the region about the y-axis.