"Integral" Example Sentences
1. The integral of a function returns the area under its graph.
2. Definite integrals are used to find the area between bounds.
3. The integral is the inverse of the derivative.
4. Multiple integration is used to calculate volumes and other properties in three dimensions.
5. The integral of sine is negative cosine.
6. Integration by parts is a useful technique for evaluating certain integrals.
7. TheFundamental Theorem of Calculus relates integrals and derivatives.
8. Improper integrals are used to find the area under curves that extend to infinity.
9. Numerical integration uses methods to approximate the value of an integral.
10. Integration allows us to find antiderivatives of functions.
11. Substitution is a common method for evaluating integrals.
12. Integral calculus is the analysis of integrals and their applications.
13. The integral from 0 to pi of sin(x) dx is equal to 2.
14. Integration and differentiation are opposite but related mathematical operations.
15. Trigonometric integrals often require the use of identities and substitutions.
16. Riemann sums are used to approximate definite integrals.
17. There are tables of integrals for commonly occurring functions.
18. Double integrals represent the volume under a surface.
19. Integrals can be evaluated using graphical and numerical methods.
20. Complex integrals are useful in applications like electric field calculations.
21. Integrals can transform differential equations into algebraic equations.
22. Triple integrals calculate volume, mass, and other physical quantities.
23. The integral of a discontinuous function must be evaluated carefully.
24. They calculated the total distance by evaluating the definite integral.
25. Students learn to solve integrals using a variety of techniques.
26. Integrals are essential for calculating areas, volumes, and other quantities.
27. SU(n) and O(n) are examples of integral Lie groups.
28. Elliptic integrals have no closed-form solution in terms of elementary functions.
29. Change of variables is used to simplify certain integrals.
30. Fourier transforms make use of integrals of complex exponential functions.
31. Contour integrals can be used to evaluate certain definite integrals.
32. Integrating factors are used to solve differential equations by integration.
33. Physics and engineering problems often involve evaluating integrals.
34. The value of an integral can often be found graphically.
35. She mastered the theory and application of integrals in calculus class.
36. Improper integrals must sometimes be evaluated using limits.
37. Percentage errors in integrals accumulate from approximation errors.
38. They found the half-life by integrating the exponential decay function.
39. Inverse trigonometric integrals require the use of identities.
40. Integration techniques include: substitution, parts, trigonometric, etc.
41. Order of integration matters when evaluating multiple integrals.
42. Improper integrals require special care at endpoints and thresholds.
43. He had to learn how to differentiate and integrate functions to pass the exam.
44. The integral was evaluated using the Fundamental Theorem of Calculus.
45. Gamma and beta integrals are examples of special functions.
46. They calculated the total distance traveled using definite integrals.
47. Residual calculus uses contour integrals to evaluate improper integrals.
48. Integration and differentiation are not truly inverse operations.
49. You can find the velocity by taking the integral of the acceleration function.
50. Irrational integrals contain irrational terms under the integral sign.
51. Probability distributions can be found by taking integrals of probability density functions.
52. They integrated the velocity function to find the total displacement.
53. Solutions to Laplace's equation are given by line integrals.
54. Green's theorem relates a surface integral to a line integral.
55. Composite functions must sometimes be broken into pieces to integrate.
56. Method of substitution is a powerful technique for evaluating integrals.
57. She studied multiple integration, line integrals and contour integrals in advanced calculus.
58. Stoke's Theorem relates volume integrals to surface integrals.
59. Integration by parts can be repeated to simplify certain integrals.
60. Students learn about the geometric interpretation of the definite integral.
Common Phases
1. An
integral part of the company's success has been customer service.
2. The
integral system failed to work properly.
3. Strong leadership has been an
integral component of the organization's growth.
4. Technology has become an
integral aspect of our daily lives.
5. The wheels are an
integral part of the bicycle.
6. Communication is
integral to any collaborative effort.
7. Teamwork was
integral to the project's success.
8. Transparency has been an
integral value of the organization.
9. Safety glasses were an
integral part of the protective gear.
10. Honesty and integrity are
integral to good leadership.
11. Problem solving skills are
integral to any job.
12. The central air conditioner is an
integral part of the home's heating and cooling system.
13. The
integral pump failed after only a few months of use.
14. Creativity is
integral to innovation and progress.
15. Research is an
integral part of the learning process.
16. The computer's motherboard is an
integral component.
17. The engine is
integral to the functioning of the car.
18. An
integral part of the process requires participants to provide feedback.
19. Ethics are
integral to good decision making.
20. The professor stressed that practice is
integral to mastery of the subject.
21. Valuing diversity has become an
integral part of the workplace culture.
22. The
integral mechanism was working properly again after repair.
23. The
integral condenser seems to have leaked all of its refrigerant.
24. The policy makers believed that infrastructure investment was
integral to economic growth.
25. Good work ethic is
integral to success in any career.
26. The
integral design featured multiple fail-safes.
27. A support system was
integral to recovery from the trauma.
28. The
integral lock was probably worn and needed replacement.
29. Efficiency is an
integral part of productivity.
30. Listening skills are
integral to effective communication.
31. Accountability has become an
integral part of the corporate culture.
32. The
integral improvements greatly enhanced the system's performance.
33. The designer believed comfort was
integral to aesthetics.
34. Hard work and persistence are
integral to achieving big goals.
35. Collaboration was
integral to the success of the interdisciplinary project.
36. Her
integral involvement helped secure funding for the program.
37. The project became
integral to the company's long-term strategy.
38. Critical thinking is an
integral skill for the 21st century workforce.
39. The
integral membrane leaked after a few years of use.
40. Quality was an
integral principle at the manufacturing company.
41. Good nutrition is
integral to a healthy lifestyle.
42. The
integral bearing assembly needed to be replaced.
43. Team values have become
integral to company culture.
44. Integrity is an
integral part of a good reputation.
45. The calculator uses the
integral to find the area under a curve.
46. The
integral part of the solution proved to be faulty.
47. Social support was
integral to her recovery and rehabilitation.
48. The fundamental design of the program proved
integral to its success.
49. An open mindset is
integral to learning and growth.
50. Proper maintenance is
integral to the longevity of any machine.
51. Logic and reason are
integral to sound decision making.
52. Flexibility is
integral to adaptability and resilience.
53. The
integral cooling system eventually failed due to lack of maintenance.
54. Self-awareness is
integral to emotional intelligence.
55. Professional development is
integral to career growth.
56. Humility is
integral to effective leadership.
57. A sense of purpose is
integral to motivation and productivity.
58. Questioning assumptions is
integral to critical thinking.
59. Integrity and character are
integral to building trust.
60. The designer believed simplicity was
integral to beauty.