An integral in calculus is a fundamental concept that deals with the accumulation of quantities, such as areas under curves, distances, or other physical quantities. There are two main types of integrals:

1. Definite Integral:

• It calculates the exact accumulation (like area) between two specific points on a function.
• It is written as: $\int_{a}^{b} f(x)\, dx$ where $a$ and $b$ are the limits of integration, $f(x)$ is the function being integrated, and $dx$ indicates that we are integrating with respect to $x$.
• The definite integral has a numerical value and represents the net area between the curve and the x-axis.

2. Indefinite Integral:

• It represents a family of functions and is the reverse process of differentiation (antiderivative).
• It is written as: $\int f(x)\, dx$ which gives a function plus a constant of integration, $C$. For example: $\int 2x\, dx = x^2 + C$

Geometrical Interpretation:

• The definite integral can be visualized as the area under the curve of the function $f(x)$ between two points on the x-axis. If the curve is above the x-axis, the area is positive; if below, the area is negative.

Fundamental Theorem of Calculus:

• This theorem links the concept of differentiation and integration. It states that if a function $F(x)$ is the antiderivative of $f(x)$, then: $\int_{a}^{b} f(x)\, dx = F(b) - F(a)$ This means that you can compute the definite integral of a function by finding its antiderivative and evaluating it at the limits $a$ and $b$.

Would you like more details or examples about integrals?

Here are 5 follow-up questions you might consider:

1. What are the practical applications of integrals in physics and engineering?
2. How do we calculate the area between two curves using integrals?
3. What are improper integrals, and how do they differ from definite integrals?
4. How do integrals apply to finding volumes of solids of revolution?
5. What are numerical methods for approximating integrals?

Tip: Always remember that integration is the reverse process of differentiation.