## Volume Between Curves Calculator

This calculator computes the volume of a solid formed by rotating the region between two curves around the x-axis.

Here's a comprehensive table summarizing all you need to know about calculating the volume between curves:

Aspect | Details |
---|---|

Definition | The volume of the solid formed by rotating the region between two curves around an axis |

General Formula | V = π ∫(a to b) [f(x)² - g(x)²] dx (for rotation around x-axis) |

Key Steps | 1. Identify the two functions f(x) and g(x) |

2. Determine the interval [a, b] | |

3. Choose the axis of rotation | |

4. Apply the appropriate formula | |

5. Integrate and evaluate | |

Rotation Around X-axis | V = π ∫(a to b) [f(x)² - g(x)²] dx |

Rotation Around Y-axis | V = 2π ∫(c to d) x[f(y) - g(y)] dy |

Washer Method | Used when rotating around x-axis or y-axis |

Shell Method | Used when rotating around y-axis or x-axis (perpendicular to the axis in the function) |

Intersection Points | Solve f(x) = g(x) to find limits of integration |

Common Mistakes | 1. Using incorrect limits of integration |

2. Forgetting to square functions for x-axis rotation | |

3. Not identifying the correct upper and lower functions | |

Tips | 1. Sketch the region to visualize the problem |

2. Check units and final answer for reasonableness | |

3. Practice with various curve combinations |

This table provides a concise overview of the key concepts, formulas, and considerations for calculating the volume between curves