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