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Finding the "area under a curve" is a way to determine how much space is enclosed between the curve of a function and the horizontal axis over an interval. In the context of definite integrals, this area is mathematically represented by the definite integral of a function over a specific range. For example, to find the area under the curve for the function \( y = t^2 \) from \( t = 1 \) to \( t = 2 \), we calculate the definite integral of \( t^2 \) with respect to \( t \).
This process involves setting up the integral with the appropriate boundaries:

• The lower limit of integration corresponds to the starting point, here \( t = 1 \).
• The upper limit of integration is the endpoint, here \( t = 2 \).

By integrating, we summarize the infinite number of infinitely small "pieces of area" beneath the curve, cumulatively resulting in the total area from \( t = 1 \) to \( t = 2 \).
Hence, the integral will give us the complete area under the curve, representing the bounded region we aim to find.

An "antiderivative" is essentially the opposite of taking a derivative. It is a function that undoes differentiation, essentially finding the original function given its derivative. In calculus, when faced with finding the area under a curve, we often need the antiderivative to perform the integration process. Suppose we have the function \( y = t^2 \). The goal is to use its antiderivative to solve the definite integral.
The antiderivative of \( t^2 \) is \( \frac{t^3}{3} \), because when you take the derivative of \( \frac{t^3}{3} \), you get back to \( t^2 \).
The antiderivative is crucial when you evaluate a definite integral, as it allows us to find the area under the curve by applying the Fundamental Theorem of Calculus:

• Compute the antiderivative.
• Evaluate it at the upper limit of integration.
• Subtract the result of evaluating it at the lower limit.

Thus, the antiderivative provides a roadmap for solving definite integrals by connecting the derivative with the original function.

"Evaluation of limits" in the context of definite integrals refers to calculating the finite area under a curve between two bounds. After finding the antiderivative of a function, the next step involves substituting the upper and lower bounds to find specific values that define this area.
Taking the function \( y = t^2 \) as an example, once the antiderivative \( \frac{t^3}{3} \) is found, the evaluation proceeds by substituting the bounds:\[ \left[ \frac{t^3}{3} \right]_1^2 \]
The upper limit \( t = 2 \) is substituted into the antiderivative to find \( \frac{8}{3} \), and the lower limit \( t = 1 \) is substituted to obtain \( \frac{1}{3} \).
The evaluation of limits translates to finding the difference between these two values, enabling us to effectively calculate the definite integral:

• Find \( \frac{8}{3} \): this is the area from 0 to \( t = 2 \).
• Find \( \frac{1}{3} \): this is the area from 0 to \( t = 1 \).
• Subtract \( \frac{1}{3} \) from \( \frac{8}{3} \): this isolates the area purely between \( t = 1 \) to \( t = 2 \).

This subtraction results in \( \frac{7}{3} \), representing the desired area of the region bounded by the curve.