Calculus I - Volumes of Solids of Revolution / Method of Rings (Practice Problems)

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Section 6.3 : Volume With Rings

For each of the following problems use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis.

Rotate the region bounded by $y = \sqrt x$ , $y = 3$ and the $y$ -axis about the $y$ -axis. Solution
Rotate the region bounded by $y = 7 - {x^2}$ , $x = - 2$ , $x = 2$ and the $x$ -axis about the $x$ -axis. Solution
Rotate the region bounded by $x = {y^2} - 6y + 10$ and $x = 5$ about the $y$ -axis. Solution
Rotate the region bounded by $y = 2{x^2}$ and $y = {x^3}$ about the $x$ -axis. Solution
Rotate the region bounded by $y = 6{{\bf{e}}^{ - 2x}}$ and $y = 6 + 4x - 2{x^2}$ between $x = 0$ and $x = 1$ about the line $y = - 2$ . Solution
Rotate the region bounded by $y = 10 - 6x + {x^2}$ , $y = - 10 + 6x - {x^2}$ , $x = 1$ and $x = 5$ about the line $y = 8$ . Solution
Rotate the region bounded by $x = {y^2} - 4$ and $x = 6 - 3y$ about the line $x = 24$ . Solution
Rotate the region bounded by $y = 2x + 1$ , $x = 4$ and $y = 3$ about the line $x = - 4$ . Solution