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Math Homework Solution

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

Completing the square gives x2 + 6x + 13 = (x + 3)2 + 22, 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 t2 - 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 x2 + 2x + 5 = (x + 1)2 + 22, 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 V0 = = 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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