Integrals example sentences

"Integrals" Example Sentences

1. To evaluate a definite integral, one must find the area under the curve between the limits of integration.
2. Improper integrals are integrals with infinite limits of integration or an integrand that is not defined at one or both of the limits of integration.
3. A definite integral is a number that represents the area between the graph of a function and the x-axis in a specific interval.
4. To calculate integrals, we use techniques such as substitution, integration by parts, and trigonometric substitution.
5. Integrals can be used in solving problems involving motion, volume, and finding averages.
6. The symbol of integration, which looks like an elongated S, represents the integration of a function.
7. The fundamental theorem of calculus provides a connection between differential calculus and integral calculus, stating that differentiation and integration are inverse operations.
8. Integrals can be used to find the total displacement of an object that has varying velocity over time.
9. In calculus, integration is the process of finding the antiderivative or indefinite integral of a function.
10. Riemann sums are used to approximate integrals, where the area under the curve is approximated by a series of rectangles.
11. The area between two curves can be found by subtracting one integral from another.
12. Trigonometric integrals can be solved using trigonometric identities or trigonometric substitution.
13. Double integrals are integrals of a function with two variables over a region in the plane.
14. Partial fractions are used to decompose a rational function into simpler fractions that can be integrated.
15. Vector calculus involves the study of vector fields and line integrals.
16. Integrals can be used to calculate the work done by a force as it moves an object.
17. Green's theorem is a way of calculating line integrals in the plane using the curl of a vector field.
18. A Laplace transform is an integral transform used to transform complex functions into simpler functions that can be more easily analyzed.
19. Integrals can be used to calculate the center of mass of an object with varying density.
20. Fourier series are used to represent a periodic function as a sum of sine and cosine terms that can be integrated.
21. The definite integral of a function is equal to the area between the function and the x-axis over a closed interval.
22. Surface integrals are used to calculate the flux of a vector field over a closed surface.
23. Integrals can be used to find the average value of a function over an interval.
24. The residue theorem is used to compute line integrals of complex functions by summing the residues of the function at its singularities.
25. Integrals can be used to model population growth, radioactive decay, and heat transfer.
26. The substitution method involves replacing the variable in an integral with a new variable in order to simplify the integral.
27. Gaussian quadrature is a numerical integration method that involves finding the roots of polynomials and using them as sample points for the integrand.
28. Integrals can be used to calculate the impulse applied to an object by a force over a time interval.
29. Contour integration is a technique used in complex analysis to evaluate line integrals of complex functions over a closed curve.
30. Differential forms are used in calculus to express integrals as a sum of products of functions and differentials.

Common Phases

1. Evaluating integrals;
2. Solving definite integrals;
3. Calculating anti-derivatives;
4. Applying integration rules;
5. Finding areas using integrals;
6. Evaluating improper integrals;
7. Using substitution method in integration;
8. Solving differential equations using integrals;
9. Understanding the fundamental theorem of calculus;
10. Applying integration in calculus and physics.