volume using cylindrical shells. We use the formula V = . Here wehave f(y) = (which we get by solving y = x^2 for x) and g(y) = 0 (the y-axis). We can easilysee that the top endpoint is y = 4 and the bottom one is y = 0. Therefore, we will find thevolume by evaluating = . We get = .
8.
When the region we are revolving is defined between a curve f(x) and g(x), we can find thevolume using cylindrical shells. We use the formula V = . Here wehave f(x) = 2 and g(x) = 0. Thus, the height of each shell is f(x) − g(x) = 2 , and theradius is simply x. We can easily see that the left endpoint is x = 0 and that the right endpoint isx = 4. Therefore, we will find the volume by evaluating = . We get = .
9.
To find the volume of a solid with a cross-section of an isosceles right triangle, we integratethe area of the square (side^2 ) over the endpoints of the interval. Here the sides of the squaresare found by f(x) − g(x) = − 0, and the intervals are found by setting y = equal to zero. We get x = −4 and x = 4. Thus, we find the volume by evaluating the integral = . We get = .