Integral Calculus

Mathematics · FE Reference Handbook section

Mathematics

2 formulas

10 exam-style examples

~49 min

All Mathematics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

The definite integral is defined as:
A table of derivatives and integrals is available in the Derivatives and Indefinite Integrals sections. The integral equations can be
used along with the following methods of integration:
C. Separation of Rational Fractions into Partial Fractions.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Definite integral — Integral Calculus

Evaluate ∫₀^2 (6x² + 3x) dx.

Given

Integrand 6x² + 3x
Limits 0 to 2

Find

The definite integral

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

Antiderivative
$F(x) = (6/3)x^{3} + (3/2)x^{2}$
Upper limit
$F(2) = (6/3)(8) + (3/2)(4)$
Evaluate
$F(2) = 16.000 + 6.000 = 22.000$
Lower limit
$F(0) = 0$
Result
$\int = 22.000$

Answer:

22.000

Why the other options are there

30.00 (integrand evaluated instead of integrated)
11.00 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 2

Definite integral — Integral Calculus (2)

Evaluate ∫₀^2 (2x² + 3x) dx.

Given

Integrand 2x² + 3x
Limits 0 to 2

Find

The definite integral

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

Antiderivative
$F(x) = (2/3)x^{3} + (3/2)x^{2}$
Upper limit
$F(2) = (2/3)(8) + (3/2)(4)$
Evaluate
$F(2) = 5.333 + 6.000 = 11.333$
Lower limit
$F(0) = 0$
Result
$\int = 11.333$

Answer:

11.333

Why the other options are there

14.00 (integrand evaluated instead of integrated)
5.67 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 3

Definite integral — Integral Calculus (3)

Evaluate ∫₀^3 (6x² + 2x) dx.

Given

Integrand 6x² + 2x
Limits 0 to 3

Find

The definite integral

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

Antiderivative
$F(x) = (6/3)x^{3} + (2/2)x^{2}$
Upper limit
$F(3) = (6/3)(27) + (2/2)(9)$
Evaluate
$F(3) = 54.000 + 9.000 = 63.000$
Lower limit
$F(0) = 0$
Result
$\int = 63.000$

Answer:

63.000

Why the other options are there

60.00 (integrand evaluated instead of integrated)
21.00 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 4

Definite integral — Integral Calculus (4)

Evaluate ∫₀^2 (2x² + 3x) dx.

Given

Integrand 2x² + 3x
Limits 0 to 2

Find

The definite integral

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

Antiderivative
$F(x) = (2/3)x^{3} + (3/2)x^{2}$
Upper limit
$F(2) = (2/3)(8) + (3/2)(4)$
Evaluate
$F(2) = 5.333 + 6.000 = 11.333$
Lower limit
$F(0) = 0$
Result
$\int = 11.333$

Answer:

11.333

Why the other options are there

14.00 (integrand evaluated instead of integrated)
5.67 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 5

Definite integral — Integral Calculus (5)

Evaluate ∫₀^4 (5x² + 1x) dx.

Given

Integrand 5x² + 1x
Limits 0 to 4

Find

The definite integral

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

Antiderivative
$F(x) = (5/3)x^{3} + (1/2)x^{2}$
Upper limit
$F(4) = (5/3)(64) + (1/2)(16)$
Evaluate
$F(4) = 106.7 + 8.000 = 114.7$
Lower limit
$F(0) = 0$
Result
$\int = 114.7$

Answer:

114.7

Why the other options are there

84.00 (integrand evaluated instead of integrated)
28.67 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 6

Definite integral — Integral Calculus (6)

Evaluate ∫₀^2 (5x² + 6x) dx.

Given

Integrand 5x² + 6x
Limits 0 to 2

Find

The definite integral

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

Antiderivative
$F(x) = (5/3)x^{3} + (6/2)x^{2}$
Upper limit
$F(2) = (5/3)(8) + (6/2)(4)$
Evaluate
$F(2) = 13.333 + 12.000 = 25.333$
Lower limit
$F(0) = 0$
Result
$\int = 25.333$

Answer:

25.333

Why the other options are there

32.00 (integrand evaluated instead of integrated)
12.67 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 7

Definite integral — Integral Calculus (7)

Evaluate ∫₀^4 (4x² + 5x) dx.

Given

Integrand 4x² + 5x
Limits 0 to 4

Find

The definite integral

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

Antiderivative
$F(x) = (4/3)x^{3} + (5/2)x^{2}$
Upper limit
$F(4) = (4/3)(64) + (5/2)(16)$
Evaluate
$F(4) = 85.333 + 40.000 = 125.3$
Lower limit
$F(0) = 0$
Result
$\int = 125.3$

Answer:

125.3

Why the other options are there

84.00 (integrand evaluated instead of integrated)
31.33 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 8

Definite integral — Integral Calculus (8)

Evaluate ∫₀^3 (5x² + 4x) dx.

Given

Integrand 5x² + 4x
Limits 0 to 3

Find

The definite integral

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

Antiderivative
$F(x) = (5/3)x^{3} + (4/2)x^{2}$
Upper limit
$F(3) = (5/3)(27) + (4/2)(9)$
Evaluate
$F(3) = 45.000 + 18.000 = 63.000$
Lower limit
$F(0) = 0$
Result
$\int = 63.000$

Answer:

63.000

Why the other options are there

57.00 (integrand evaluated instead of integrated)
21.00 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 9

Definite integral — Integral Calculus (9)

Evaluate ∫₀^2 (5x² + 1x) dx.

Given

Integrand 5x² + 1x
Limits 0 to 2

Find

The definite integral

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

Antiderivative
$F(x) = (5/3)x^{3} + (1/2)x^{2}$
Upper limit
$F(2) = (5/3)(8) + (1/2)(4)$
Evaluate
$F(2) = 13.333 + 2.000 = 15.333$
Lower limit
$F(0) = 0$
Result
$\int = 15.333$

Answer:

15.333

Why the other options are there

22.00 (integrand evaluated instead of integrated)
7.67 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus

Example 10

Definite integral — Integral Calculus (10)

Evaluate ∫₀^4 (4x² + 4x) dx.

Given

Integrand 4x² + 4x
Limits 0 to 4

Find

The definite integral

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

Antiderivative
$F(x) = (4/3)x^{3} + (4/2)x^{2}$
Upper limit
$F(4) = (4/3)(64) + (4/2)(16)$
Evaluate
$F(4) = 85.333 + 32.000 = 117.3$
Lower limit
$F(0) = 0$
Result
$\int = 117.3$

Answer:

117.3

Why the other options are there

80.00 (integrand evaluated instead of integrated)
29.33 (average value reported)

Reference: FE Reference Handbook — Mathematics → Integral Calculus