## Ex 2: find the area under a curve using a geometric formula

There has always been a constant term left when integrating. As a consequence, such integrals are referred to as infinite integrals. We integrate a function between two points with definite integrals, and we can find the precise value of the integral without any uncertain constant terms [the constant cancels out].
Areas below the x-axis will be negative, while those above it will be positive. This means you must be cautious when looking for an area that is partially above and partially below the x-axis.

## How to find the area of a shaded region under a curve

However, there were some limitations to this description. We needed to be non-negative and constant. Unfortunately, real-world issues do not always adhere to these guidelines. We’ll look at how to use the definite integral to apply the definition of the region under the curve to a broader range of functions in this section.
The integral symbol used in the preceding description should be familiar to you. Similar notation was used to describe an antiderivative in the chapter on Applications of Derivatives, where we used the infinite integral symbol (without the and above and below). While the notation for indefinite integrals and definite integrals which appear to be identical, they are not the same. A number is a definite integral. A family of functions is referred to as an infinite integral. We’ll look at how these ideas are connected later in this chapter. However, we must always pay careful attention to the notation to determine if we are dealing with a definite or indefinite integral.

### Grade 3 math #11.2, square units and area

A probability distribution with a probability density function is known as a continuous probability distribution. Since its cumulative distribution function is totally continuous with respect to the Lebesgue measure [latex]lambda[/latex], mathematicians call such a distribution “absolutely continuous.” When the distribution of [latex]textX[/latex] is continuous, it is referred to as a continuous random variable. Continuous probability distributions come in a variety of shapes and sizes, including regular, uniform, chi-squared, and others.
In comparison to a discrete distribution, where the set of possible values for the random variable is at most countable, a continuous random variable may take a continuous range of values. Because an occurrence with probability zero is impossible for a discrete distribution (for example, rolling 3 and a half on a standard die is impossible and has probability zero), this is not the case for a continuous random variable.
When measuring the width of an oak leaf, for example, a result of 3.5 cm is possible; however, it has a likelihood of zero since there are uncountably many other possible values even between 3 cm and 4 cm. Although each of these outcomes has a probability of zero, the probability that the outcome will fall within the interval (3 cm, 4 cm) is nonzero. Given that the likelihood that [latex]textX[/latex] attains any value within an infinite set, such as an interval, cannot be identified by naively adding the probabilities for individual values, this obvious paradox is resolved. In mathematical terms, each value has an infinitesimally small probability, which is statistically equal to zero.