How do you do these questions?

Step-by-step explanation:

The width of each interval is Δx = (4−1)/6 = 1/2.

a) Evaluate the function at the beginning and end of each interval.

f(x) = 7√(ln x)

f(1) = 7√(ln 1) = 0

f(1.5) = 7√(ln 1.5) ≈ 4.4573300

f(2) = 7√(ln 2) ≈ 5.8278823

f(2.5) = 7√(ln 2.5) ≈ 6.7006153

f(3) = 7√(ln 3) ≈ 7.3370295

f(3.5) = 7√(ln 3.5) ≈ 7.8348826

f(4) = 7√(ln 4) ≈ 8.2418702

Calculate the area of each trapezoid.

T₁ = ½ (0 + 4.4573300) (1/2) = 1.1143325

T₂ = ½ (4.4573300 + 5.8278823) (1/2) = 2.5713031

T₃ = ½ (5.8278823 + 6.7006153) (1/2) = 3.1321244

T₄ = ½ (6.7006153 + 7.3370295) (1/2) = 3.5094112

T₅ = ½ (7.3370295 + 7.8348826) (1/2) = 3.7929780

T₆ = ½ (7.8348826 + 8.2418702) (1/2) = 4.0191882

The total area is therefore:

T = 1.1143325 + 2.5713031 + 3.1321244 + 3.5094112 + 3.7929780 + 4.0191882

T = 18.139337

b) Evaluate the function at the midpoint of each interval.

f(1.25) = 7√(ln 1.25) ≈ 3.3066651

f(1.75) = 7√(ln 1.75) ≈ 5.2365230

f(2.25) = 7√(ln 2.25) ≈ 6.3036165

f(2.75) = 7√(ln 2.75) ≈ 7.0404861

f(3.25) = 7√(ln 3.25) ≈ 7.5996115

f(3.75) = 7√(ln 3.75) ≈ 8.0477348

Calculate the area of each rectangle.

M₁ = (3.3066651) (1/2) = 1.6533325

M₂ = (5.2365230) (1/2) = 2.6182615

M₃ = (6.3036165) (1/2) = 3.1518082

M₄ = (7.0404861) (1/2) = 3.5202431

M₅ = (7.5996115) (1/2) = 3.7998057

M₆ = (8.0477348) (1/2) = 4.0238674

The total area is therefore:

M = 1.6533325 + 2.6182615 + 3.1518082 + 3.5202431 + 3.7998057 + 4.0238674

M = 18.767318

c) Simpson's rule can be calculated as:

S₆ = Δx/3 [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + 4f(x₅) + f(x₆)]

S₆ = (1/2)/3 [0 + 4(4.4573300) + 2(5.8278823) + 4(6.7006153) + 2(7.3370295) + 4(7.8348826) + 8.2418702]

S₆ = 18.423834

points are handed out whenever you post answers.

brainliest answers are handed out by the person asking the question, so whether your answer is picked depends entirely on whether the asker bothers to pick

if you're asking about your rank, it depends on having a certain number of points and a certain number of brainliest answers. for example, at the time of this writing, i have 44338 points out of 15000 needed for the next (nearly triple the requirement! ) but i only have 24 out of 50 required brainliest answers. this number decays, meaning after a certain amount of time (i think about a month), the total number of brainliest answers you've earned doesn't count toward this total. (i've been stuck at the same rank for a long

the lesson to learn here is that you shouldn't worry about points or rank. chances are, no matter how good an answer you may provide, your efforts will go mostly unnoticed unless you focus on the newest/most active questions that get posted.