Volterra integral equations of the first kind are mathematical equations that involve an unknown function, which is integrated over a specific interval. They are typically expressed in the form f(t) = \int_a^b K(t, s) \phi(s) \, ds , where f(t) is a known function, K(t, s) is a kernel function, and \phi(s) is the unknown function to be determined. These equations are named after the Italian mathematician Vito Volterra, who contributed significantly to the field of integral equations. Volterra integral equations of the first kind are often used in various applications, including physics, engineering, and biology, to model systems where the current state depends on past states.