Problem 12 - Applied Calculus

Question

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{1}^{3} 6 x^{2} d x$$

Step-by-Step Solution

Answer

The integral evaluates to 52.

Step 1: Identify the Integral

The expression you need to integrate is \(\int_{1}^{3} 6x^2 \, dx\). This is a definite integral with limits of integration from 1 to 3.

Step 2: Find the Antiderivative

To integrate \(6x^2\), find the antiderivative. The antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). Apply this rule to \(6x^2\):\[\int 6x^2 \, dx = 6 \cdot \frac{x^{3}}{3} + C = 2x^3 + C\] where \(C\) is the constant of integration.

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, for a function \(F(x)\) which is the antiderivative of \(f(x)\), the definite integral from \(a\) to \(b\) is given by:\[ F(b) - F(a) \] Substitute \(F(x) = 2x^3\), \(a = 1\), and \(b = 3\):\[ F(3) - F(1) = 2(3)^3 - 2(1)^3 \]

Step 4: Evaluate the Expression

Calculate \(2(3)^3\) and \(2(1)^3\):\[ 2(3)^3 = 2 \cdot 27 = 54 \]\[ 2(1)^3 = 2 \cdot 1 = 2 \]Subtract to find the value of the definite integral:\[ 54 - 2 = 52 \]

Key Concepts

Definite Integrals

Definite integrals are an important concept in calculus, offering a way to calculate the area under a curve within specific limits. When you see the expression \( \int_{a}^{b} f(x) \, dx \), it means you are calculating the integral of the function \( f(x) \) from point \( a \) to point \( b \). This mathematical operation provides the net area between the function and the x-axis, taking into account that areas below the x-axis are considered negative.

• The "limits of integration," \( a \) and \( b \), are critical as they define the interval over which you integrate.
• A definite integral results in a numerical value, not a function, as it measures a finite area.

The Fundamental Theorem of Calculus connects definite integrals and antiderivatives, simplifying this calculation by finding the antiderivative \( F(x) \) and evaluating \( F(b) - F(a) \). So, when evaluating \( \int_{1}^{3} 6x^2 \, dx \) in your exercise, you consider the difference in the function's antiderivative at the upper limit minus the lower limit. This practical application shows how definite integrals give concise solutions to geometric and real-world problems.

Antiderivatives

An antiderivative of a function is essentially its "reverse derivative." While a derivative provides the rate of change, the antiderivative helps to reconstruct the original function from its rate of change. When a function \( f(x) \) is given, finding \( F(x) \) such that \( F'(x) = f(x) \) is finding its antiderivative.For example, the antiderivative of \( 6x^2 \) was calculated using the power rule for integration. The power rule states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), which led to the solution \( \int 6x^2 \, dx = 2x^3 + C \). Here:

• \( 6x^2 \) is the function we integrate.
• The factor of 6 is multiplied after finding the antiderivative of \( x^2 \), resulting in \( 2x^3 + C \).
• \( C \) represents any constant, since antiderivatives differ by a constant.

Antiderivatives are essential not just for calculating integrals, but also for solving differential equations, often representing functions in a more complete form than just their rates of change alone.

Integration Techniques

Understanding and using the right integration techniques is crucial to solving calculus problems effectively. In the context of the Fundamental Theorem of Calculus, it's vital to choose the correct method to find the integral’s antiderivative. Here are some techniques often used:

• **Basic Power Rule**: Useful when integrating polynomials like \( x^n \). As in the example, \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \).
• **Substitution**: Used when a function can be simplified into a basic form, making it easier to integrate.
• **Integration by Parts**: Applies when two functions are multiplied and neither is easy to integrate alone.

Selecting the right technique saves time and simplifies complex integrals. For \( \int_{1}^{3} 6x^2 \, dx \), a straightforward application of the power rule was sufficient to find the antiderivative: \( 2x^3 + C \). This illustrates how understanding various techniques allows for flexibility in problem-solving, enabling a more profound approach to diverse calculus problems.