Which one of the following is a correct method for defining and initializing an integer variable with name value?

The choice that defines and initializes an integer variable correctly is presented in the first option. In Java, the programming language often used in introductory programming courses like CSE110, variable names must begin with a letter and can include digits, underscores, and dollar signs, accepting varied casing, such as lowercase 'int.'

Furthermore, 'int' is the proper keyword to declare an integer type variable. When defining a variable, it's essential to follow the syntax of specifying the data type (in this case, 'int'), the variable name ('value'), an assignment operator ('='), and a valid integer value (30). The number 30 is a valid integer literal.

The other choices provide variations that do not conform to Java's syntax rules. For example, capitalizing 'Int' fails since Java is case-sensitive, treating 'Int' as an undefined type. Additionally, using '.30' in options C and D defines a decimal (floating-point) number, which does not align with the integer data type specified by 'int'. Thus, the first option is the only viable answer that follows the correct syntax for declaring and initializing an integer variable in Java.