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The definite integral is a fundamental concept in calculus that quantifies the net area under a function's graph between two specific points. Imagine sketching a curve on a graph; the definite integral calculates the total 'accumulated' area between the curve and the x-axis, from the lower limit to the upper limit.

For instance, when we encounter an expression like \(\int_{a}^{b} f(t) dt\), it means we're computing the definite integral of the function \(f(t)\) from \(t = a\) to \(t = b\). This process becomes especially interesting when 'a' and 'b' are not constants, but variables themselves, leading us to functions defined by integrals, like \(f(x, y)=\int_{x}^{y} \sqrt{1+t^{3}} d t\), as seen in our original problem.

Understanding the properties of definite integrals, such as linearity, additivity over intervals, and the behavior under variable substitution, is essential in solving problems that involve these expressions and their derivatives.

The Fundamental Theorem of Calculus (FTC) is the bridge that connects differentiation and integration, two core operations in calculus. It's divided into two parts. The first part guarantees that if a function is integrable over \[a, b\] and has an antiderivative, then it can be evaluated using this antiderivative. The second part, which is crucial to our exercise, tells us how to differentiate an integral function.

The essence of the second part is that the derivative of the integral of a function is the original function itself. If we have \(F(x) = \int_{a}^{x} f(t) dt\), then \(F'(x) = f(x)\), given that \(f\) is continuous at \(x\). So, in the context of our problem with \(f(x, y)\), we apply the FTC to find its partial derivatives, leading us through the path to discover \(f_{x}(x, y)\) and \(f_{y}(x, y)\) by considering the impact of each variable independently.

Multivariable calculus extends the principles of single-variable calculus to functions of several variables. It's the realm where we examine surfaces and contours instead of just curves, and we consider partial derivatives and multiple integrals when we want to understand changes and areas.

In the problem presented, we encounter \(f(x, y)\) which is a function of two variables. The partial derivatives, \(f_{x}(x, y)\) and \(f_{y}(x, y)\), represent the rate of change of \(f\) with respect to one variable while holding the other constant. Addressing partial derivatives is essential in multivariable functions since changes aren't uniform across all directions, which is a fundamental divergence from single-variable calculus.

An interesting point about our function defined as a definite integral is that it illustrates a specific link between multivariable calculus and integral calculus—the latter being a subset of the former when functions of several variables are considered. This exemplifies the beauty of calculus; how different concepts are interconnected and build on each other, moving from the simple to the complex.