MathsAges 16-18

Definite power-rule integrals

Integrate a single power term and evaluate the definite integral between x = 0 and an upper bound. Review suitability for your learners before teaching.

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Worksheet StudioPractice pack · Ages 16-18

Definite power-rule integrals

Take your time. Show your thinking.

1. Evaluate the definite integral of 12x^3 from x = 0 to x = 5.
12 · 3 · 0 · 5 · Definite power-rule integr…
2. Evaluate the definite integral of 9x^2 from x = 0 to x = 2.
9 · 2 · 0 · 2 · Definite power-rule integr…
3. Evaluate the definite integral of 16x^3 from x = 0 to x = 3.
16 · 3 · 0 · 3 · Definite power-rule integr…
4. Evaluate the definite integral of 8x^3 from x = 0 to x = 1.
8 · 3 · 0 · 1 · Definite power-rule integr…
5. Evaluate the definite integral of 28x^3 from x = 0 to x = 6.
28 · 3 · 0 · 6 · Definite power-rule integr…
6. Evaluate the definite integral of 20x^3 from x = 0 to x = 2.
20 · 3 · 0 · 2 · Definite power-rule integr…
7. Evaluate the definite integral of 10x^1 from x = 0 to x = 3.
10 · 1 · 0 · 3 · Definite power-rule integr…
8. Evaluate the definite integral of 3x^2 from x = 0 to x = 1.
3 · 2 · 0 · 1 · Definite power-rule integr…
9. Evaluate the definite integral of 12x^1 from x = 0 to x = 1.
12 · 1 · 0 · 1 · Definite power-rule integr…
10. Evaluate the definite integral of 2x^1 from x = 0 to x = 1.
2 · 1 · 0 · 1 · Definite power-rule integr…
11. Evaluate the definite integral of 4x^1 from x = 0 to x = 3.
4 · 1 · 0 · 3 · Definite power-rule integr…
12. Evaluate the definite integral of 6x^2 from x = 0 to x = 2.
6 · 2 · 0 · 2 · Definite power-rule integr…

12 questions · Seed 51 · Age guidance, not curriculum certification

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Worksheet StudioAnswer key · Ages 16-18

Definite power-rule integrals

Answers and reasoning. Review before teaching.

1. Evaluate the definite integral of 12x^3 from x = 0 to x = 5.
1875
An antiderivative of 12x^3 is 3x^4. Evaluating from 0 to 5 gives 3 × 5^4 = 1875.
2. Evaluate the definite integral of 9x^2 from x = 0 to x = 2.
24
An antiderivative of 9x^2 is 3x^3. Evaluating from 0 to 2 gives 3 × 2^3 = 24.
3. Evaluate the definite integral of 16x^3 from x = 0 to x = 3.
324
An antiderivative of 16x^3 is 4x^4. Evaluating from 0 to 3 gives 4 × 3^4 = 324.
4. Evaluate the definite integral of 8x^3 from x = 0 to x = 1.
2
An antiderivative of 8x^3 is 2x^4. Evaluating from 0 to 1 gives 2 × 1^4 = 2.
5. Evaluate the definite integral of 28x^3 from x = 0 to x = 6.
9072
An antiderivative of 28x^3 is 7x^4. Evaluating from 0 to 6 gives 7 × 6^4 = 9072.
6. Evaluate the definite integral of 20x^3 from x = 0 to x = 2.
80
An antiderivative of 20x^3 is 5x^4. Evaluating from 0 to 2 gives 5 × 2^4 = 80.
7. Evaluate the definite integral of 10x^1 from x = 0 to x = 3.
45
An antiderivative of 10x^1 is 5x^2. Evaluating from 0 to 3 gives 5 × 3^2 = 45.
8. Evaluate the definite integral of 3x^2 from x = 0 to x = 1.
1
An antiderivative of 3x^2 is 1x^3. Evaluating from 0 to 1 gives 1 × 1^3 = 1.
9. Evaluate the definite integral of 12x^1 from x = 0 to x = 1.
6
An antiderivative of 12x^1 is 6x^2. Evaluating from 0 to 1 gives 6 × 1^2 = 6.
10. Evaluate the definite integral of 2x^1 from x = 0 to x = 1.
1
An antiderivative of 2x^1 is 1x^2. Evaluating from 0 to 1 gives 1 × 1^2 = 1.
11. Evaluate the definite integral of 4x^1 from x = 0 to x = 3.
18
An antiderivative of 4x^1 is 2x^2. Evaluating from 0 to 3 gives 2 × 3^2 = 18.
12. Evaluate the definite integral of 6x^2 from x = 0 to x = 2.
16
An antiderivative of 6x^2 is 2x^3. Evaluating from 0 to 2 gives 2 × 2^3 = 16.

12 questions · Seed 51 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Definite power-rule integrals

Take your time. Show your thinking.

1. Evaluate the definite integral of 12x^3 from x = 0 to x = 5.
12 · 3 · 0 · 5 · Definite power-rule integr…
2. Evaluate the definite integral of 9x^2 from x = 0 to x = 2.
9 · 2 · 0 · 2 · Definite power-rule integr…
3. Evaluate the definite integral of 16x^3 from x = 0 to x = 3.
16 · 3 · 0 · 3 · Definite power-rule integr…
4. Evaluate the definite integral of 8x^3 from x = 0 to x = 1.
8 · 3 · 0 · 1 · Definite power-rule integr…
5. Evaluate the definite integral of 28x^3 from x = 0 to x = 6.
28 · 3 · 0 · 6 · Definite power-rule integr…
6. Evaluate the definite integral of 20x^3 from x = 0 to x = 2.
20 · 3 · 0 · 2 · Definite power-rule integr…
7. Evaluate the definite integral of 10x^1 from x = 0 to x = 3.
10 · 1 · 0 · 3 · Definite power-rule integr…
8. Evaluate the definite integral of 3x^2 from x = 0 to x = 1.
3 · 2 · 0 · 1 · Definite power-rule integr…
9. Evaluate the definite integral of 12x^1 from x = 0 to x = 1.
12 · 1 · 0 · 1 · Definite power-rule integr…
10. Evaluate the definite integral of 2x^1 from x = 0 to x = 1.
2 · 1 · 0 · 1 · Definite power-rule integr…
11. Evaluate the definite integral of 4x^1 from x = 0 to x = 3.
4 · 1 · 0 · 3 · Definite power-rule integr…
12. Evaluate the definite integral of 6x^2 from x = 0 to x = 2.
6 · 2 · 0 · 2 · Definite power-rule integr…

12 questions · Seed 51 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Definite power-rule integrals

Answers and reasoning. Review before teaching.

1. Evaluate the definite integral of 12x^3 from x = 0 to x = 5.
1875
An antiderivative of 12x^3 is 3x^4. Evaluating from 0 to 5 gives 3 × 5^4 = 1875.
2. Evaluate the definite integral of 9x^2 from x = 0 to x = 2.
24
An antiderivative of 9x^2 is 3x^3. Evaluating from 0 to 2 gives 3 × 2^3 = 24.
3. Evaluate the definite integral of 16x^3 from x = 0 to x = 3.
324
An antiderivative of 16x^3 is 4x^4. Evaluating from 0 to 3 gives 4 × 3^4 = 324.
4. Evaluate the definite integral of 8x^3 from x = 0 to x = 1.
2
An antiderivative of 8x^3 is 2x^4. Evaluating from 0 to 1 gives 2 × 1^4 = 2.
5. Evaluate the definite integral of 28x^3 from x = 0 to x = 6.
9072
An antiderivative of 28x^3 is 7x^4. Evaluating from 0 to 6 gives 7 × 6^4 = 9072.
6. Evaluate the definite integral of 20x^3 from x = 0 to x = 2.
80
An antiderivative of 20x^3 is 5x^4. Evaluating from 0 to 2 gives 5 × 2^4 = 80.
7. Evaluate the definite integral of 10x^1 from x = 0 to x = 3.
45
An antiderivative of 10x^1 is 5x^2. Evaluating from 0 to 3 gives 5 × 3^2 = 45.
8. Evaluate the definite integral of 3x^2 from x = 0 to x = 1.
1
An antiderivative of 3x^2 is 1x^3. Evaluating from 0 to 1 gives 1 × 1^3 = 1.
9. Evaluate the definite integral of 12x^1 from x = 0 to x = 1.
6
An antiderivative of 12x^1 is 6x^2. Evaluating from 0 to 1 gives 6 × 1^2 = 6.
10. Evaluate the definite integral of 2x^1 from x = 0 to x = 1.
1
An antiderivative of 2x^1 is 1x^2. Evaluating from 0 to 1 gives 1 × 1^2 = 1.
11. Evaluate the definite integral of 4x^1 from x = 0 to x = 3.
18
An antiderivative of 4x^1 is 2x^2. Evaluating from 0 to 3 gives 2 × 3^2 = 18.
12. Evaluate the definite integral of 6x^2 from x = 0 to x = 2.
16
An antiderivative of 6x^2 is 2x^3. Evaluating from 0 to 2 gives 2 × 2^3 = 16.

12 questions · Seed 51 · Age guidance, not curriculum certification