An engineer applies the trapezoidal rule to approximate the area under a velocity curve. Given step size (h) = 1.4000; f(x0) (f0) = 7.4000; f(x1) (f1) = 5.7000; f(x2) (f2) = 5.9000, determine the approx. integral (I).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for I:

   $$I = \dfrac{h}{2}\left[f_0 + 2f_1 + f_2\right]$$

3. Step 3

   $$List the givens: step size (h) = 1.4000, f(x_{0}) (f_{0}) = 7.4000, f(x_{1}) (f_{1}) = 5.7000, f(x_{2}) (f_{2}) = 5.9000$$

4. Step 4 — Substitute the given values:

   $$I = \dfrac{1.4000}{2}\left[f_0 + 2f_1 + f_2\right]$$

5. Step 5 — Evaluate:

   $$I = 17.2900$$

6. Step 6 — Check: returning I = 17.2900 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 34.5800 — kept a factor of two that cancels in the correct rearrangement.
• 8.6450 — dropped that same factor in the other direction.
• 19.0190 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

A student uses the trapezoidal rule to estimate an integral from tabulated data. Given f(x0) (f0) = 4.8000; f(x1) (f1) = 1.0000; f(x2) (f2) = 3.0000; approx. integral (I) = 100.8, determine the step size (h).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except h is given, so isolate h symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for h:

   $$h = \dfrac{I}{(f_0+2f_1+f_2)/2}$$

3. Step 3

   $$List the givens: f(x_{0}) (f_{0}) = 4.8000, f(x_{1}) (f_{1}) = 1.0000, f(x_{2}) (f_{2}) = 3.0000, approx. integral (I) = 100.8$$

4. Step 4 — Substitute the given values:

   $$h = \dfrac{100.8}{(f_0+2f_1+f_2)/2}$$

5. Step 5 — Evaluate:

   $$h = 20.5714$$

6. Step 6 — Check: returning h = 20.5714 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 41.1429 — kept a factor of two that cancels in the correct rearrangement.
• 10.2857 — dropped that same factor in the other direction.
• 22.6286 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

The trapezoidal rule is used to approximate the flow volume from measured readings. Given step size (h) = 2.8000; f(x0) (f0) = 4.4000; f(x2) (f2) = 2.0000; approx. integral (I) = 47.7300, determine the f(x1) (f1).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except f1 is given, so isolate f1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for f1:

   $$f_{1} = \dfrac{\frac{2I}{h} - f_0 - f_2}{2}$$

3. Step 3 — List the givens: step size (h) = 2.8000, f(x0) (f0) = 4.4000, f(x2) (f2) = 2.0000, approx. integral (I) = 47.7300.

4. Step 4 — Substitute the given values:

   $$f_{1} = \dfrac{\frac{247.7300}{2.8000} - f_0 - f_2}{2}$$

5. Step 5 — Evaluate:

   $$f_{1} = 13.8464$$

6. Step 6 — Check: returning f1 = 13.8464 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 27.6929 — kept a factor of two that cancels in the correct rearrangement.
• 6.9232 — dropped that same factor in the other direction.
• 15.2311 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

An engineer applies the trapezoidal rule to approximate the area under a velocity curve. Given step size (h) = 1.5000; f(x0) (f0) = 3.9000; f(x1) (f1) = 4.9000; f(x2) (f2) = 3.2000, determine the approx. integral (I).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for I:

   $$I = \dfrac{h}{2}\left[f_0 + 2f_1 + f_2\right]$$

3. Step 3

   $$List the givens: step size (h) = 1.5000, f(x_{0}) (f_{0}) = 3.9000, f(x_{1}) (f_{1}) = 4.9000, f(x_{2}) (f_{2}) = 3.2000$$

4. Step 4 — Substitute the given values:

   $$I = \dfrac{1.5000}{2}\left[f_0 + 2f_1 + f_2\right]$$

5. Step 5 — Evaluate:

   $$I = 12.6750$$

6. Step 6 — Check: returning I = 12.6750 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 25.3500 — kept a factor of two that cancels in the correct rearrangement.
• 6.3375 — dropped that same factor in the other direction.
• 13.9425 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

A student uses the trapezoidal rule to estimate an integral from tabulated data. Given f(x0) (f0) = 8.6000; f(x1) (f1) = 5.3000; f(x2) (f2) = 7.4000; approx. integral (I) = 53.3900, determine the step size (h).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except h is given, so isolate h symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for h:

   $$h = \dfrac{I}{(f_0+2f_1+f_2)/2}$$

3. Step 3

   $$List the givens: f(x_{0}) (f_{0}) = 8.6000, f(x_{1}) (f_{1}) = 5.3000, f(x_{2}) (f_{2}) = 7.4000, approx. integral (I) = 53.3900$$

4. Step 4 — Substitute the given values:

   $$h = \dfrac{53.3900}{(f_0+2f_1+f_2)/2}$$

5. Step 5 — Evaluate:

   $$h = 4.0143$$

6. Step 6 — Check: returning h = 4.0143 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 8.0286 — kept a factor of two that cancels in the correct rearrangement.
• 2.0071 — dropped that same factor in the other direction.
• 4.4157 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

The trapezoidal rule is used to approximate the flow volume from measured readings. Given step size (h) = 1.5000; f(x0) (f0) = 5.6000; f(x2) (f2) = 2.7000; approx. integral (I) = 84.2500, determine the f(x1) (f1).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except f1 is given, so isolate f1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for f1:

   $$f_{1} = \dfrac{\frac{2I}{h} - f_0 - f_2}{2}$$

3. Step 3 — List the givens: step size (h) = 1.5000, f(x0) (f0) = 5.6000, f(x2) (f2) = 2.7000, approx. integral (I) = 84.2500.

4. Step 4 — Substitute the given values:

   $$f_{1} = \dfrac{\frac{284.2500}{1.5000} - f_0 - f_2}{2}$$

5. Step 5 — Evaluate:

   $$f_{1} = 52.0167$$

6. Step 6 — Check: returning f1 = 52.0167 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 104.0 — kept a factor of two that cancels in the correct rearrangement.
• 26.0083 — dropped that same factor in the other direction.
• 57.2183 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

An engineer applies the trapezoidal rule to approximate the area under a velocity curve. Given step size (h) = 0.8000; f(x0) (f0) = 5.7000; f(x1) (f1) = 1.5000; f(x2) (f2) = 4.4000, determine the approx. integral (I).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for I:

   $$I = \dfrac{h}{2}\left[f_0 + 2f_1 + f_2\right]$$

3. Step 3

   $$List the givens: step size (h) = 0.8000, f(x_{0}) (f_{0}) = 5.7000, f(x_{1}) (f_{1}) = 1.5000, f(x_{2}) (f_{2}) = 4.4000$$

4. Step 4 — Substitute the given values:

   $$I = \dfrac{0.8000}{2}\left[f_0 + 2f_1 + f_2\right]$$

5. Step 5 — Evaluate:

   $$I = 5.2400$$

6. Step 6 — Check: returning I = 5.2400 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 10.4800 — kept a factor of two that cancels in the correct rearrangement.
• 2.6200 — dropped that same factor in the other direction.
• 5.7640 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

A student uses the trapezoidal rule to estimate an integral from tabulated data. Given f(x0) (f0) = 6.8000; f(x1) (f1) = 4.1000; f(x2) (f2) = 5.9000; approx. integral (I) = 82.6200, determine the step size (h).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except h is given, so isolate h symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for h:

   $$h = \dfrac{I}{(f_0+2f_1+f_2)/2}$$

3. Step 3

   $$List the givens: f(x_{0}) (f_{0}) = 6.8000, f(x_{1}) (f_{1}) = 4.1000, f(x_{2}) (f_{2}) = 5.9000, approx. integral (I) = 82.6200$$

4. Step 4 — Substitute the given values:

   $$h = \dfrac{82.6200}{(f_0+2f_1+f_2)/2}$$

5. Step 5 — Evaluate:

   $$h = 7.9062$$

6. Step 6 — Check: returning h = 7.9062 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 15.8124 — kept a factor of two that cancels in the correct rearrangement.
• 3.9531 — dropped that same factor in the other direction.
• 8.6968 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

The trapezoidal rule is used to approximate the flow volume from measured readings. Given step size (h) = 2.4000; f(x0) (f0) = 7.1000; f(x2) (f2) = 9.7000; approx. integral (I) = 59.9200, determine the f(x1) (f1).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except f1 is given, so isolate f1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for f1:

   $$f_{1} = \dfrac{\frac{2I}{h} - f_0 - f_2}{2}$$

3. Step 3 — List the givens: step size (h) = 2.4000, f(x0) (f0) = 7.1000, f(x2) (f2) = 9.7000, approx. integral (I) = 59.9200.

4. Step 4 — Substitute the given values:

   $$f_{1} = \dfrac{\frac{259.9200}{2.4000} - f_0 - f_2}{2}$$

5. Step 5 — Evaluate:

   $$f_{1} = 16.5667$$

6. Step 6 — Check: returning f1 = 16.5667 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 33.1333 — kept a factor of two that cancels in the correct rearrangement.
• 8.2833 — dropped that same factor in the other direction.
• 18.2233 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule

An engineer applies the trapezoidal rule to approximate the area under a velocity curve. Given step size (h) = 1.8000; f(x0) (f0) = 6.2000; f(x1) (f1) = 1.2000; f(x2) (f2) = 7.7000, determine the approx. integral (I).

Start with the thinking

• The governing relation printed in this handbook section is Trapezoidal rule for numerical integration.
• Everything except I is given, so isolate I symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trapezoidal rule approximates a definite integral by summing trapezoid areas between panel points.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

2. Step 2 — Rearrange symbolically for I:

   $$I = \dfrac{h}{2}\left[f_0 + 2f_1 + f_2\right]$$

3. Step 3

   $$List the givens: step size (h) = 1.8000, f(x_{0}) (f_{0}) = 6.2000, f(x_{1}) (f_{1}) = 1.2000, f(x_{2}) (f_{2}) = 7.7000$$

4. Step 4 — Substitute the given values:

   $$I = \dfrac{1.8000}{2}\left[f_0 + 2f_1 + f_2\right]$$

5. Step 5 — Evaluate:

   $$I = 14.6700$$

6. Step 6 — Check: returning I = 14.6700 to

   $$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0) + 2f(x_1) + f(x_2)\right]$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 29.3400 — kept a factor of two that cancels in the correct rearrangement.
• 7.3350 — dropped that same factor in the other direction.
• 16.1370 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trapezoidal Rule