Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{1 / 2}^{1}\left(t^{-3}-8\right)$$

Short answer

Answer: The value of the definite integral is $$-\frac{5}{2}$$.

Step-by-step solution

Find the antiderivative

To find the antiderivative, integrate each term of the function with respect to t: $$\int\left(t^{-3}-8\right)dt = \int t^{-3} dt - \int 8 dt$$ Now let's find the antiderivative of each term separately: $$\int t^{-3} dt = \frac{t^{-2}}{-2} + C_1$$ $$\int 8 dt = 8t + C_2$$ Now, combine the terms to find the antiderivative of the whole function: $$\int\left(t^{-3}-8\right)dt = \frac{t^{-2}}{-2} - 8t + C$$

Apply the Fundamental Theorem of Calculus

Now that we have found the antiderivative of the function, we can apply the Fundamental Theorem of Calculus to find the definite integral: $$\int_{1/2}^{1}\left(t^{-3}-8\right)dt = \left[\frac{t^{-2}}{-2} - 8t\right]_{1/2}^1$$ Evaluate the antiderivative at the limits of the integral: $$F(1) = \frac{1^{-2}}{-2} - 8(1) = -\frac{1}{2} - 8 = -\frac{17}{2}$$ $$F\left(\frac{1}{2}\right) = \frac{\left(\frac{1}{2}\right)^{-2}}{-2} - 8\left(\frac{1}{2}\right) = -2 - 4 = -6$$ Now, apply the Fundamental Theorem of Calculus: $$\int_{1/2}^{1}\left(t^{-3}-8\right)dt = F(1) - F\left(\frac{1}{2}\right) = -\frac{17}{2} - (-6) = -\frac{17}{2} + 6 = -\frac{5}{2}$$

The integral value

The definite integral is: $$\int_{1/2}^{1}\left(t^{-3}-8\right)dt = -\frac{5}{2}$$

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus that quantify the net area under a curve between two points on the x-axis. They're expressed in the form \[ \int_{a}^{b} f(x) \,dx \], where \(a\) and \(b\) are the lower and upper limits of integration, respectively, and \(f(x)\) is the function being integrated. A definite integral calculates the accumulated value of \(f(x)\) over the interval \[a, b\].

When working with definite integrals, it's essential to remember that they are not just about area, but can represent a variety of physical and mathematical concepts depending on the context, such as displacement in physics, or the total accumulation of a quantity over time. Each definite integral has a value that includes the function's behavior within the specified limits, making it a powerful tool for solving real-world problems.

Antiderivative Calculation

An antiderivative of a function \(f\) is another function \(F\) whose derivative is \(f\). In other words, it's the original function from which \(f\) has been derived through differentiation. Calculating an antiderivative, a process also referred to as integration, is often the first step in evaluating definite integrals using the Fundamental Theorem of Calculus.

To calculate an antiderivative, you need to apply the reverse rules of differentiation. For instance, the antiderivative of \(t^{-3}\) is \(\frac{t^{-2}}{-2} + C_1\), and the antiderivative of a constant, such as 8, is \(8t + C_2\), where \(C_1\) and \(C_2\) are constants of integration. In the context of definite integrals, these arbitrary constants cancel out, as you're ultimately concerned with the change in \(F\) over the interval.

Integral Evaluation

Integral evaluation is the final part of solving a definite integral problem. Once an antiderivative is found, it must be evaluated at both the upper and lower limits of integration. The Fundamental Theorem of Calculus parts the process into two primary operations: finding the antiderivative and evaluating it at the limits of integration to compute the integral's value.

Evaluating the antiderivative \(F\) at the upper bound \(b\) gives \(F(b)\), and at the lower bound \(a\) gives \(F(a)\). The definite integral's value is the difference \(F(b) - F(a)\). So, when you're given the integral \[\int_{1/2}^{1}\left(t^{-3}-8\right)dt\], you calculate the value of the integrated function at \(t = 1\) and \(t = 1/2\), then subtract to find that the area under the curve from \(t = 1/2\) to \(t = 1\) is \( -\frac{5}{2}\). This final step solidifies the abstract concept of integration into a concrete value that can be applied in numerous scenarios.