Concepts to remember on Definite Integral

1. Definite integral as the limit of a sum:

• The definite integral of a function f(x) over an interval [a, b] is defined as the limit of a sum of areas of rectangles, as the number of rectangles approaches infinity and their widths approach zero.

2. The fundamental theorem of calculus:

• The fundamental theorem of calculus states that the integral of a continuous function f(x) over an interval [a, b] can be calculated as F(b) - F(a), where F(x) is any antiderivative of f(x).

3. Integration formulas for basic functions:

• Remember the integration formulas for basic functions, such as the power rule, logarithmic rule, exponential rule, and trigonometric rules.

4. Riemann sums and their properties:

• Understand the concept of Riemann sums as approximations of definite integrals and know the properties of Riemann sums, such as upper and lower sums and their relationship to the definite integral.

5. Properties of definite integrals, including linearity, additivity, and the mean value theorem:

• Definite integrals have properties like linearity (i.e., integral of a sum equals the sum of integrals), additivity (i.e., integral of a function over two intervals equals the sum of integrals), and the mean value theorem (i.e., there exists a point in the interval where the function takes the average value).

6. Applications of definite integrals, such as finding areas, volumes, and arc lengths:

• Grasp how to use definite integrals to calculate areas of regions bounded by curves, volumes of solids generated by revolving or cross-sectioning regions, and arc lengths of curves.

7. Improper integrals and their convergence or divergence:

• Understand the concept of improper integrals, where the limits of integration are infinite or the function has discontinuities, and learn how to determine their convergence or divergence using comparison or limit comparison tests.

8. Comparison and limit comparison tests for improper integrals:

• Be familiar with the comparison and limit comparison tests for determining the convergence or divergence of improper integrals.

9. Differentiation and integration of definite integrals:

• Learn how to differentiate and integrate definite integrals, including the use of the second fundamental theorem of calculus.

10. Techniques of integration, such as u-substitution, integration by parts, and trigonometric substitution:

• Master various techniques of integration, such as u-substitution (change of variable), integration by parts, and trigonometric substitution, to evaluate integrals.

11. Applications of definite integrals in physics, engineering, economics, and other fields:

• Explore the applications of definite integrals in various fields, such as physics (e.g., work and energy), engineering (e.g., fluid mechanics), economics (e.g., consumer surplus and producer surplus), and other disciplines.

Remember that practice is essential to solidify your understanding of these concepts. Work through practice problems and examples, and seek help when needed.