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The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.

The graph of the function \(y = f(x)\) is shown above. What is the value of \[\int_0^6 f(x) dx?\]

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Steve knows that if he uses right-hand endpoints with 2 subintervals to find a Reimann sum approximation of \[\int_0^8 x dx,\] his approximation will be the sum of the areas of the 2 rectangles shown. Assuming all his work is correct, what will his approximation be?

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Morgan and Pat use right-hand endpoints to find Reimann sum approximations of \[\int_0^8 x dx.\]

If Morgan uses 2 subintervals, and Pat uses 4 subintervals, whose approximation is closer to the true value of the integral?

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Ferb knows that \[\int_0^8 x dx = 32,\] and is interested in how close a right-hand Reimann sum approximation to the integral can be.

Based on his work so far, if Ferb uses 32 subintervals, what will his approximation be?

Number of subintervals | Approximation | Error |

1 | 64 | 32 |

2 | 48 | 16 |

4 | 40 | 8 |

8 | 36 | 4 |

16 | 34 | 2 |

32 | ?? |

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\[ A = \int_{1}^{2} x dx, \,\,\,\, B = \int_{1}^{2} x^2 dx \]

Which of the following is true of \(A\) and \(B?\)

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