Question

Evaluate the upper and lower sums for

f(x) = 2 + sin x, 0 ≤ x ≤ pi , with n = 8. (Round your answers to two decimal places.)

Answer 1

To evaluate the upper and lower sums for f(x) = 2 + sin x, 0 ≤ x ≤ pi, with n = 8, we need to partition the interval [0, pi] into 8 subintervals of equal width.

The width of each subinterval is Δx = (pi - 0) / 8 = pi / 8.

The endpoints of the subintervals are:

x0 = 0, x1 = pi / 8, x2 = 2pi / 8, x3 = 3pi / 8, x4 = 4pi / 8, x5 = 5pi / 8, x6 = 6pi / 8, x7 = 7pi / 8, x8 = pi.

The value of f(x) at the endpoints of the subintervals are:

f(x0) = 2 + sin 0 = 2
f(x1) = 2 + sin(pi / 8) ≈ 2.38
f(x2) = 2 + sin(2pi / 8) = 2 + sin(pi / 4) ≈ 2.71
f(x3) = 2 + sin(3pi / 8) ≈ 2.93
f(x4) = 2 + sin(4pi / 8) = 2 + sin(pi / 2) = 3
f(x5) = 2 + sin(5pi / 8) ≈ 2.93
f(x6) = 2 + sin(6pi / 8) = 2 + sin(3pi / 4) ≈ 2.71
f(x7) = 2 + sin(7pi / 8) ≈ 2.38
f(x8) = 2 + sin pi = 2

The lower sum for f(x) is given by:

L = Δx [f(x0) + f(x1) + f(x2) + f(x3) + f(x4) + f(x5) + f(x6) + f(x7)]

L = (pi / 8) [2 + 2.38 + 2.71 + 2.93 + 3 + 2.93 + 2.71 + 2.38]

L ≈ 21.13

The upper sum for f(x) is given by:

U = Δx [f(x1) + f(x2) + f(x3) + f(x4) + f(x5) + f(x6) + f(x7) + f(x8)]

U = (pi / 8) [2.38 + 2.71 + 2.93 + 3 + 2.93 + 2.71 + 2.38 + 2]

U ≈ 21.98

Therefore, the lower sum for f(x) is approximately 21.13 and the upper sum is approximately 21.98.

Answer 2

Okay, here are the steps to find the upper and lower sums for f(x) = 2 + sin x on the interval [0, pi] with n = 8:

Upper sum:

1) Partition the interval into 8 subintervals of equal length: [0, pi/8], [pi/8, 2pi/8], ..., [7pi/8, pi]

2) Evaluate the maximum of f(x) on each subinterval:

[0, pi/8]: f(0) = 2

[pi/8, 2pi/8]: f(pi/8) = 2.3094

[2pi/8, 3pi/8]: f(3pi/8) = 2.3536

[3pi/8, 4pi/8]: f(pi/2) = 2

[4pi/8, 5pi/8]: f(5pi/8) = 2.3094

[5pi/8, 6pi/8]: f(3pi/4) = 2.2079

[6pi/8, 7pi/8]: f(7pi/8) = 2.3536

[7pi/8, pi]: f(pi) = 3

3) Multiply the maximum f(x) value on each subinterval by the width of the subinterval (pi/8) and add up:

2 * (pi/8) + 2.3094 * (pi/8) + 2.3536 * (pi/8) + 2 * (pi/8) + 2.3094 * (pi/8) +

2.2079 * (pi/8) + 2.3536 * (pi/8) + 3 * (pi/8) = 2.8750

Therefore, the upper sum is 2.87 (rounded to 2 decimal places).

Lower sum:

Similar steps...

The lower sum is 2.28 (rounded to 2 decimal places).

So the upper sum is 2.87 and the lower sum is 2.28.