# Math Homework Solution

7.5, #46. The integrand is an improper rational function . Long division gives

Completing the square gives x^{2} + 6x + 13 = (x +
3)^{2} + 2^{2}, so set w = x + 3 and dw = dx.

Then

so formula 25 applies with a = 2, b = 6, c = -5, giving

7.5, #50. The integrand is an improper rational function . Long division gives

Since t^{2} - 1 = (t - 1)(t + 1), formula 26 with
a = 1, b = -1 gives

7.5, #51. Formula 24 with a = 2 gives

The left and right Riemann sums with n = 100 are
approximately 0.3939 and 0.3915. Since

the integrand is decreasing on the interval [0, 2], its value is between these
two estimates.

7.5, #53. Since x^{2} + 2x + 5 = (x + 1)^{2}
+ 2^{2}, formula 24 with a = 2 gives

The left and right Riemann sums with n = 100 are
approximately 0.1613 and 0.1605. Since

the integrand is decreasing on the interval [0, 1], its value is between these
two estimates .

7.5, #64. (a) The average voltage over a second is

(b) The average of V ^{2} over a second is

using the substitution u = 120πt and du = 120πdt. Thus

(c) If = 110, then V_{0}
= = 155.56 volts.

7.6, #1.

n = 1 | n = 2 | n = 4 | |

LEFT | 40.0000 | 40.7846 | 41.7116 |

RIGHT | 51.2250 | 46.3971 | 44.5179 |

TRAP | 45.6125 | 43.5909 | 43.1147 |

MID | 41.5692 | 42.6386 | 42.8795 |

7.6, #3.

n | 10 | 100 | 1000 |

LEFT | 5.4711 | 5.8116 | 5.8464 |

RIGHT | 6.2443 | 5.8890 | 5.8541 |

TRAP | 5.8577 | 5.8503 | 5.8502 |

MID | 5.8465 | 5.8502 | 5.8502 |

is increasing and concave-up on [1, 2] since

are positive on [1, 2]. (
and for )
**So LEFT** and** MID**

underestimate the integral, **RIGHT** and **TRAP** overestimate the
integral

7.6, #4.

n | 20 | 100 | 1000 |

LEFT | 3.0132 | 2.9948 | 2.9930 |

RIGHT | 2.9711 | 2.9906 | 2.9925 |

TRAP | 2.9922 | 2.9927 | 2.9927 |

MID | 2.9930 | 2.9927 | 2.9927 |

is decreasing and concave-down on [0, ] since

are negative on (The
second derivative is messy, but its sign is easy to determine.

The numerator is always negative since square roots are non-negative; the
denominator

is positive for ) **So LEFT** and **MID**
overestimate the integral,** RIGHT** and

**TRAP** underestimate the integral.

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