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course Mth 174

7.57.5.1

Because the function is increasing and concave upward, LEFT(2)

7.5.2

Because the function is decreasing and concave downward, RIGHT(2)< int(f(x) dx a,b)

7.5.3

Because the function is increasing and concave downward, LEFT(2)

7.5.4

Because the function is decreasing and concave upward, RIGHT(2)< int(f(x) dx a,b)

7.5.7

For n=2 subdivisions of the interval [0,6], we use delta x = 3

LEFT(2) = f(0)(3) + f(3)(3) = 27

RIGHT(2) = f(3)(3) + f(6)(3) = 27 + 108 = 135

TRAP(2) = (f(0)+f(3)) (3) (1/2) + (f(3)+f(6)) (3) (1/2) = 81/2 + 135/2 = 81

MID(2) = f(1.5)(3) + f(4.5)(3) = 6.75 + 60.75 = 67.5

7.5.10

For n = 2 subdivisions of interval [0, `pi], we use delta x = `pi/2

LEFT(2) = f(0)( `pi/2) + f( `pi/2) ( `pi/2) = 1.571

RIGHT(2) = f( `pi/2) ( `pi/2) + f(`pi) ( `pi/2) = 1.571

TRAP(2) = [f(0) + f( `pi/2)]/2 ( `pi/2) + [f( `pi/2) + f(`pi)]/2 ( `pi/2) = 1.571

MID(2) = f(`pi/4) ( `pi/2) + f(3`pi/4) ( `pi/2) = 2.221

7.5.15

Because the function is increasing on the interval, RIGHT(n) is guaranteed give an overestimate and LEFT(n) give an underestimate.

7.5.18

Because the function is concave upward on the interval, TRAP(n) is guaranteed give an overestimate and MID(n) give an underestimate.

7.5.21

a. Int(sinx dx, 0, 2`pi) = -cos(2`pi) + cos(0) = 0

b. for n = 1, delta x =2 `pi, midpoint is x = `pi, in which sin(`pi) = 0

for n = 2, delta x = `pi, midpoint is x = `pi/2, in which sin(`pi/2) = -sin(3`pi/4)

c. for n = 3, delta x = 2`pi/3

MID(3) = f(`pi/3)(2`pi/3) + f(`pi)(2`pi/3) + f(5`pi/3) (2`pi/3) = 0

7.5.22

a. when x = 1, y = `sqrt(2-1^2) = 1

arctan(1/1) = `pi/4

area = 1*1/2 + `pi(sqrt(2)^2)(`pi/4)/2`pi = 1/2 + `pi/4

b. for n = 5, we use delta x = 0.2

LEFT(5) = f(0)(0.2) + f(0.2)(0.2) + f(0.4)(0.2) + f(0.6)(0.2) + f(0.8)(0.2) = 1.324

RIGHT(5) = f(0.2)(0.2) + f(0.4)(0.2) + f(0.6)(0.2) + f(0.8)(0.2) + f(1)(0.2) = 1.241

TRAP(5) = [f(0) + f(0.2)](0.1) + [f(0.2) +f(0.4)](0.1) + [f(0.4) + f(0.6)](0.1) + [f(0.6) + f(0.8)](0.1) + [f(0.8) + f(1)](0.1) = 1.284

MID(5) = f(0.1)0.2 + f(0.3)0.2 + f(0.5)0.2 + f(0.7)0.2 + f(0.9)0.2 = 1.287