Evaluate ∫₀^5 (5x² + 1x) dx.

Start with the thinking

• Antidifferentiate term by term, then apply the limits.
• The lower limit of zero kills the second evaluation.

Step-by-step solution

1. Antiderivative

   $$F(x) = (5/3)x^{3} + (1/2)x^{2}$$

2. Upper limit

   $$F(5) = (5/3)(125) + (1/2)(25)$$

3. Evaluate

   $$F(5) = 208.3 + 12.500 = 220.8$$

4. Lower limit

   $$F(0) = 0$$

5. Result

   $$\int = 220.8$$

Why the other options are there

• 130.0 (integrand evaluated instead of integrated)
• 44.17 (average value reported)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Use Newton's algorithm on f(x) = x² − 8 with a starting value x₀ = 2 to obtain two improved estimates of √8, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(2 + 8/2) = 3.00000$$

3. Iteration 2

   $$x_{2} = ½(3.00000 + 8/3.00000) = 2.83333$$

4. Exact value — √8 = 2.82843

5. Error — |x₂ − √8| = 0.004906

Why the other options are there

• 4.0000 (single division, no averaging)
• 6.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Estimate ∫₀^6 (x² + 1) dx with the trapezoidal rule using n = 4 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (6 - 0)/4 = 1.5000$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 34.5000, f(6) = 37.000

4. Substituting

   $$I \approx (1.5000/2)[107.0] = 80.2500$$

5. Exact

   $$\int = b^{3}/3 + b = 78.0000$$

6. Error

   $$|I - exact|/exact = 2.885$$

Why the other options are there

• 160.5 (dropped the ½)
• 28.500 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Evaluate ∫₀^2 (6x² + 2x) dx.

Start with the thinking

• Antidifferentiate term by term, then apply the limits.
• The lower limit of zero kills the second evaluation.

Step-by-step solution

1. Antiderivative

   $$F(x) = (6/3)x^{3} + (2/2)x^{2}$$

2. Upper limit

   $$F(2) = (6/3)(8) + (2/2)(4)$$

3. Evaluate

   $$F(2) = 16.000 + 4.000 = 20.000$$

4. Lower limit

   $$F(0) = 0$$

5. Result

   $$\int = 20.000$$

Why the other options are there

• 28.00 (integrand evaluated instead of integrated)
• 10.00 (average value reported)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Use Newton's algorithm on f(x) = x² − 8 with a starting value x₀ = 5 to obtain two improved estimates of √8, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(5 + 8/5) = 3.30000$$

3. Iteration 2

   $$x_{2} = ½(3.30000 + 8/3.30000) = 2.86212$$

4. Exact value — √8 = 2.82843

5. Error — |x₂ − √8| = 0.033694

Why the other options are there

• 1.6000 (single division, no averaging)
• -12.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Estimate ∫₀^7 (x² + 1) dx with the trapezoidal rule using n = 4 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (7 - 0)/4 = 1.7500$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 45.8750, f(7) = 50.000

4. Substituting

   $$I \approx (1.7500/2)[142.8] = 124.9$$

5. Exact

   $$\int = b^{3}/3 + b = 121.3$$

6. Error

   $$|I - exact|/exact = 2.945$$

Why the other options are there

• 249.8 (dropped the ½)
• 44.625 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Evaluate ∫₀^5 (6x² + 5x) dx.

Start with the thinking

• Antidifferentiate term by term, then apply the limits.
• The lower limit of zero kills the second evaluation.

Step-by-step solution

1. Antiderivative

   $$F(x) = (6/3)x^{3} + (5/2)x^{2}$$

2. Upper limit

   $$F(5) = (6/3)(125) + (5/2)(25)$$

3. Evaluate

   $$F(5) = 250.0 + 62.500 = 312.5$$

4. Lower limit

   $$F(0) = 0$$

5. Result

   $$\int = 312.5$$

Why the other options are there

• 175.0 (integrand evaluated instead of integrated)
• 62.50 (average value reported)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Use Newton's algorithm on f(x) = x² − 6 with a starting value x₀ = 4 to obtain two improved estimates of √6, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(4 + 6/4) = 2.75000$$

3. Iteration 2

   $$x_{2} = ½(2.75000 + 6/2.75000) = 2.46591$$

4. Exact value — √6 = 2.44949

5. Error — |x₂ − √6| = 0.016419

Why the other options are there

• 1.5000 (single division, no averaging)
• -6.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Estimate ∫₀^6 (x² + 1) dx with the trapezoidal rule using n = 8 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (6 - 0)/8 = 0.7500$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 85.7500, f(6) = 37.000

4. Substituting

   $$I \approx (0.7500/2)[209.5] = 78.5625$$

5. Exact

   $$\int = b^{3}/3 + b = 78.0000$$

6. Error

   $$|I - exact|/exact = 0.721$$

Why the other options are there

• 157.1 (dropped the ½)
• 14.250 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Numerical Integration

Evaluate ∫₀^4 (2x² + 5x) dx.

Start with the thinking

• Antidifferentiate term by term, then apply the limits.
• The lower limit of zero kills the second evaluation.

Step-by-step solution

1. Antiderivative

   $$F(x) = (2/3)x^{3} + (5/2)x^{2}$$

2. Upper limit

   $$F(4) = (2/3)(64) + (5/2)(16)$$

3. Evaluate

   $$F(4) = 42.667 + 40.000 = 82.667$$

4. Lower limit

   $$F(0) = 0$$

5. Result

   $$\int = 82.667$$

Why the other options are there

• 52.00 (integrand evaluated instead of integrated)
• 20.67 (average value reported)

Reference: FE Reference Handbook — Mathematics → Numerical Integration