Hello my dear lovely fellow math help forum users!!
Can you please check if I have done these Calculus problems correctly or not? I have taken a picture of the problems with my work in writing. You can enlarge the image with this link: http://img62.imageshack.us/img62/323/123niz.jpg
Please reply if you notice where I have made a mistake... Thanks guys, Much Love
He used the "shell" method. Imagine a vertical line at a fixed from (x, 0) to . Rotated around the y- axis, that creates a hollow cylinder. The length of the line is and it is rotated in a circle of radius x so circumference . The surface area of that cylinder is . Now if that cylinder has "thickness" dx, the volume of the cylinder is . Integrating that from 0 to 2 gives the volume: .
nuvydeep, your original work for the first problem was correct. Since you had reversed the order of the endpoints, you did not need the negative sign in the front.
However, in your haste, you may have written the final answer wrong. The final expression should be . Notice that , so the final answer is simply .