Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{4} t(t-2)(t-4) d t$$

Short answer

Answer: The value of the definite integral is 0.

Step-by-step solution

Find the antiderivative of the integrand

To find the antiderivative of \(t(t-2)(t-4)\), first expand the polynomial: \(t(t-2)(t-4) = t^3 - 6t^2 + 8t\) Now, using the power rule, find the antiderivative: \(F(t) = \frac{1}{4}t^4 - 2t^3 + 4t^2 + C\)

Use the Fundamental Theorem of Calculus to evaluate the definite integral

Now, applying the Fundamental Theorem of Calculus to evaluate the definite integral between 0 and 4: $$\int_{0}^{4} t(t-2)(t-4) dt = F(4) - F(0)$$ First, estimate the integral at the upper limit: \(F(4) = \frac{1}{4}(4)^4 - 2(4)^3 + 4(4)^2 = 64 - 128 + 64 = 0\) Next, estimate the integral at the lower limit: \(F(0) = \frac{1}{4}(0)^4 - 2(0)^3 + 4(0)^2 = 0\) Finally, subtract F(0) from F(4): $$\int_{0}^{4} t(t-2)(t-4) dt = F(4) - F(0) = 0 - 0 = 0$$

Key concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is an essential principle that connects differentiation and integration, two core concepts in calculus. It has two main parts, and the one usually relevant to solving definite integrals is the second part. This theorem states that if you have a continuous function on a closed interval \[a, b\], and F is the antiderivative of f on this interval, then the definite integral of f from a to b is given by \(F(b) - F(a)\).

When tackling problems with definite integrals, such as \(\int_{0}^{4} t(t-2)(t-4) dt\), we use the Fundamental Theorem of Calculus to simplify our work. Instead of adding up an infinite number of tiny areas under the curve, which is what definite integration conceptually is, we merely evaluate the antiderivative of the function at the upper and lower limits of the interval and subtract the two. This process transforms a potentially complex problem into a much simpler calculation of functions values at specific points.

Antiderivative

An antiderivative is a function whose derivative is the original function we started with. This concept is crucial when working with integrals because it helps us reverse the process of differentiation. To solve a definite integral, we first need to find the antiderivative of the integrand, the mathematical function that is being integrated.

Looking at our example, \(\int_{0}^{4} t(t-2)(t-4) dt\), the first step towards the solution is to find the antiderivative of the integrand \(t(t-2)(t-4)\). It’s important to remember that, unlike differentiation, anti-differentiation, or finding antiderivatives, is not a straight-forward process and often requires some algebraic manipulation, such as expanding polynomials as shown in the step-by-step solution.

Power Rule

The power rule is one of the most fundamental and widely used techniques for finding antiderivatives in calculus. It is based on a relatively simple formula that states if \(f(x) = x^n\), where \(n\) is any real number except -1, then the antiderivative of \(f\) is \(F(x) = \frac{1}{n+1}x^{n+1}\) plus a constant of integration C.

Applying this rule in our exercise, after expanding the polynomial \(t(t-2)(t-4)\), we identified the terms that resemble the power function \(x^n\). By increasing the exponent by 1 and dividing by the new exponent, we found the antiderivative \(F(t) = \frac{1}{4}t^4 - 2t^3 + 4t^2 + C\). It is important to note that we include a 'C' for the general antiderivative, but it cancels out when evaluating a definite integral, as subtraction eliminates it from the final result.