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.

7.6, #3.
 

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.

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.