Verify That Fx Is An Antiderivative Of The Integrand Fx

Verify That Fx Is An Antiderivative Of The Integrand Fx And Use

Verify that F(x) is an antiderivative of the integrand f(x) and use Part 2 of the Fundamental Theorem to evaluate the definite integrals. Use substitution to find an antiderivative in terms of u, then express the answer back in terms of x. Apply the change of variable technique to find an antiderivative in terms of x. Evaluate the definite integrals. Sketch the graphs of each function and determine the area between the graphs of f and g over the specified interval.

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The process of verifying that a given function \(F(x)\) is an antiderivative of a function \(f(x)\) is fundamental in calculus, especially for evaluating definite integrals using the Fundamental Theorem of Calculus. This theorem states that if \(F(x)\) is an antiderivative of \(f(x)\) on \([a, b]\), then the definite integral from \(a\) to \(b\) of \(f(x)\) is given by \(F(b) - F(a)\). Therefore, the initial step involves confirming that the derivative \(F'(x) = f(x)\). Once established, this relationship allows us to evaluate definite integrals efficiently without explicitly integrating \(f(x)\) repeatedly.

Suppose \(F(x)\) is provided, and the integrand \(f(x)\) is known. To verify this, differentiate \(F(x)\) and compare the result to \(f(x)\). If they are equal for all \(x\) in the interval, then \(F(x)\) is indeed an antiderivative of \(f(x)\). This verification is essential because it underpins the application of the Fundamental Theorem of Calculus, facilitating the evaluation of definite integrals directly from known antiderivatives.

Once the antiderivative \(F(x)\) is verified, the next step involves computing specific definite integrals. Since the theorem states that \(\int_a^b f(x) dx = F(b) - F(a)\), evaluating the integral reduces to calculating \(F(b)\) and \(F(a)\). This approach simplifies the integral computation and provides an exact value, which is particularly useful in applications such as calculating areas under curves, displacement, and accumulated quantities.

In some cases, the integrand \(f(x)\) can be complicated, and a substitution method helps simplify the integration process. Typically, substitution involves identifying a part of the integrand to replace with a new variable \(u\). For example, if \(f(x)\) contains a composite function like \(g(h(x))\), choosing \(u = h(x)\) allows rewriting the integral in terms of \(u\) and \(du\). Computing \(du\) involves differentiating \(u\) with respect to \(x\), i.e., \(du = h'(x) dx\). This substitution transforms the integral into a simpler form, often reducing it to a basic power rule integral or a known standard form.

After rewriting the integral in terms of \(u\), one finds an antiderivative in the \(u\)-variable. The result is then expressed back in terms of \(x\) by substituting \(u = h(x)\) once the integration is complete. This method not only eases the computation but also deepens understanding of the relationships within the integral through variable change, which is a core concept in methods like substitution and variable change techniques in calculus.

The change of variables technique extends beyond substitution; it involves systematically replacing parts of the integral to simplify computation, especially for more complex functions. When applying this method, it is essential to accurately compute \(du\) and change the limits of integration accordingly if working with a definite integral. This ensures the correctness of the evaluated integral and maintains the integrity of the original problem.

Evaluating definite integrals after performing substitution involves calculating the antiderivative in terms of \(u\), then substituting the upper and lower bounds expressed in terms of \(u\). Alternatively, one can substitute within the integral before evaluating. The result provides the numerical value of the area under the curve between specific bounds, which is often the ultimate goal in applications involving accumulated quantities or graphical analysis.

Graphical representation plays a pivotal role in understanding integrals. Sketching the functions \(f(x)\) and \(g(x)\) over the interval of interest helps visualize the area between the curves. The bounded region's area corresponds to the absolute value of the integral of their difference \(|f(x) - g(x)|\) over the interval. Accurately sketching the graphs provides insights into relative maxima, minima, points of intersection, and how the functions compare across the domain.

In conclusion, verifying antiderivatives, employing substitution and change of variable techniques, evaluating definite integrals, and visualizing functions through graphs are fundamental skills in calculus. These methods streamline complex computations, deepen understanding of the integral's geometric interpretation, and are essential tools in solving a wide range of applied problems in science and engineering.

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