Riemann Sums & Definite Integrals Worksheet 9

Compute exact integrals and sums, distinguish area from signed integrals, and include constants for indefinite integrals.

hard: Use another set to practice explaining and checking your answers.

18 problems · 25-30 minutes · Skills: integrals, Riemann sums, calculus

About This Worksheet

This hard-level worksheet contains 18 problems designed for 12th Grade students practicing riemann sums & definite integrals. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Manipulate algebraic expressions and understand functions, limits and derivative notation.

Compute exact integrals and sums, distinguish area from signed integrals, and include constants for indefinite integrals.
Check your result using the method in the example before moving to the next question.

How to Solve: Worked Example

Example Problem: Approximate the area under f(x) = 3x on [0, 2] using 2 equal rectangles and right endpoints.

Solution Steps:

The rectangle width is 1.
Right endpoints are 1, 2, …, n. Add their function values and multiply by the width.

Answer: 9 square units

Tip: Check your result using the method in the example before moving to the next question.

Paper Include separate answer key in PDF

Approximate the area under f(x) = 2x on [0, 2] using 2 equal rectangles and right endpoints.

Approximate the area under f(x) = 3x on [0, 6] using 6 equal rectangles and right endpoints.

Approximate the area under f(x) = 1x on [0, 6] using 6 equal rectangles and right endpoints.

Approximate the area under f(x) = 3x on [0, 5] using 5 equal rectangles and right endpoints.

Approximate the area under f(x) = 1x on [0, 5] using 5 equal rectangles and right endpoints.

Approximate the area under f(x) = 5x on [0, 6] using 6 equal rectangles and right endpoints.

Approximate the area under f(x) = 1x on [0, 4] using 4 equal rectangles and right endpoints.

Approximate the area under f(x) = 5x on [0, 4] using 4 equal rectangles and right endpoints.

Approximate the area under f(x) = 2x on [0, 6] using 6 equal rectangles and right endpoints.

Approximate the area under f(x) = 4x on [0, 2] using 2 equal rectangles and right endpoints.

Approximate the area under f(x) = 6x on [0, 6] using 6 equal rectangles and right endpoints.

Approximate the area under f(x) = 3x on [0, 3] using 3 equal rectangles and right endpoints.

Approximate the area under f(x) = 1x on [0, 3] using 3 equal rectangles and right endpoints.

Approximate the area under f(x) = 2x on [0, 4] using 4 equal rectangles and right endpoints.

Approximate the area under f(x) = 3x on [0, 2] using 2 equal rectangles and right endpoints.

Approximate the area under f(x) = 5x on [0, 2] using 2 equal rectangles and right endpoints.

Approximate the area under f(x) = 4x on [0, 3] using 3 equal rectangles and right endpoints.

Approximate the area under f(x) = 5x on [0, 3] using 3 equal rectangles and right endpoints.

Answer Key

1. 6 square units2. 63 square units3. 21 square units4. 45 square units5. 15 square units6. 105 square units7. 10 square units8. 50 square units9. 42 square units10. 12 square units11. 126 square units12. 18 square units13. 6 square units14. 20 square units15. 9 square units16. 15 square units17. 24 square units18. 30 square units

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Frequently Asked Questions

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What skills does this integrals worksheet cover?

This worksheet focuses on integrals, Riemann sums, calculus. It is designed for 12th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 18 problems. It typically takes 25-30 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.