Final answer:
To find the volume formed by rotating a region enclosed by a curve around a line, use the method of cylindrical shells and the formula V = 2π∫(x)(f(x)) dx. Determine the limits of integration by finding the points of intersection between the curve and the line of rotation. Calculate the volume by evaluating the integral.
Step-by-step explanation:
In order to find the volume formed by rotating a region enclosed by a curve around a line, we can use the method of cylindrical shells. The formula to calculate the volume using cylindrical shells is V = 2π∫(x)(f(x)) dx, where x represents the variable of integration and f(x) represents the function that forms the curve.
To find the limits of integration, we need to determine the points where the curve intersects the line of rotation. Then, we integrate the function from the lower bound to the upper bound to calculate the volume.
For example, if we rotate the region enclosed by the curve y = x^2 and the line x = 1 around the y-axis, we can set up the integral as follows:
V = 2π∫(x)(x^2) dx from x = 0 to x = 1
Simplifying the integral and evaluating it will give us the volume of the region.