Definite integrals Evaluate the following integrals using...

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.
$$\int_{0}^{5}\left(x^{2}-9\right) d x$$

Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities, typically interpreted as the net area under a curve between two specified limits along the x-axis. It is a fundamental tool in calculus used to calculate areas, volumes, and other quantities derived from continuous functions.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration by stating that if a function is continuous over an interval, then its definite integral over that interval can be evaluated using any of its antiderivatives. Specifically, it shows that the net accumulation is the difference between the values of an antiderivative at the upper and lower limits.

Antiderivative

An antiderivative of a function is another function whose derivative is equal to the original function. In the context of definite integrals, finding an antiderivative is a key step as it allows one to compute the net accumulated value by applying the Fundamental Theorem of Calculus.

Net Area (Signed Area)

The term 'net area' refers to the result of the definite integral, which accounts for areas above and below the x-axis with a sign convention; areas above the axis contribute positively, while areas below contribute negatively. This concept is crucial when interpreting the physical meaning of integrals, particularly in applications where the direction of accumulation matters.

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