Problem 50 - Applied Calculus

Question

Evaluate each definite integral. $$ \int_{-2}^{2}\left(3 w^{2}-2 w\right) d w $$

Step-by-Step Solution

Answer

The definite integral evaluates to 16.

Step 1: Find the Antiderivative

To evaluate the definite integral, we first find the antiderivative of the integrand. The integrand here is a polynomial: \( 3w^2 - 2w \). We find the antiderivative by using the power rule, which states that the antiderivative of \( w^n \) is \( \frac{w^{n+1}}{n+1} \). Thus,\[\int (3w^2 - 2w) \, dw = \frac{3w^3}{3} - \frac{2w^2}{2} + C = w^3 - w^2 + C\]where \( C \) is the constant of integration.

Step 2: Apply the Limits of Integration

Now that we have the antiderivative \( w^3 - w^2 \), we apply the limits to calculate the definite integral. The limits given are \(-2\) to \(2\). According to the Fundamental Theorem of Calculus, we calculate \( F(2) - F(-2) \) where \( F(w) = w^3 - w^2 \).Evaluate \( F(2) \):\[ F(2) = 2^3 - 2^2 = 8 - 4 = 4 \]Evaluate \( F(-2) \):\[ F(-2) = (-2)^3 - (-2)^2 = -8 - 4 = -12 \].

Step 3: Compute the Definite Integral

Finally, we find the value of the definite integral by subtracting \( F(-2) \) from \( F(2) \).\[F(2) - F(-2) = 4 - (-12) = 4 + 12 = 16\]Thus, the value of the definite integral is \( 16 \).

Key Concepts

Antiderivative

An antiderivative is a function whose derivative gives back the original function. It is often also known as the indefinite integral. If you have a function, say \( f(w) \), an antiderivative of this function, expressed as \( F(w) \), should satisfy the condition that when you differentiate \( F(w) \), you return to the function \( f(w) \). In mathematical terms, \( F'(w) = f(w) \).

In practice, to evaluate definite integrals, we first need to find an antiderivative. For instance, when dealing with the polynomial \( 3w^2 - 2w \), we need to find a function \( F(w) \) such that the derivative of \( F(w) \) yields the polynomial itself. In this case, the antiderivative of \( 3w^2 \) is \( w^3 \), and the antiderivative of \( -2w \) is \( -w^2 \). Hence, the combined antiderivative is \( w^3 - w^2 + C \), where \( C \) is the constant of integration that is not needed for definite integrals.

Power Rule

The power rule is a straightforward technique used to find the antiderivative of polynomial functions. It states that to find the antiderivative of a function \( w^n \), you increase the exponent by one and then divide by this new exponent. Mathematically, this can be expressed as:

• Antiderivative of \( w^n = \frac{w^{n+1}}{n+1} + C \)

Applying the power rule simplifies the process of finding an antiderivative. When using it for the polynomial \( 3w^2 - 2w \), you consider each term separately. For \( 3w^2 \), you apply the power rule, resulting in \( \frac{3w^3}{3} \), which simplifies to \( w^3 \). Similarly, for \( -2w \), the antiderivative is \( \frac{-2w^2}{2} \), simplifying to \( -w^2 \).

This step is crucial in integral calculus, as it allows you to handle each term of a polynomial individually, making the process of integration systematic and error-free.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a key principle connecting differentiation and integration, two main concepts in calculus. It essentially establishes that integration can be reversed by differentiation and vice versa. The theorem consists of two main parts:

• The first part states that if you have a continuous function \( f(w) \), and \( F(w) \) is its antiderivative, then the definite integral of \( f(w) \) from \( a \) to \( b \) can be calculated using \( F \).
• The second part tells us that \( F(b) - F(a) \) gives the net area under the curve of \( f(w) \) between \( a \) and \( b \).

To apply this to our example, once we've found that \( F(w) = w^3 - w^2 \), the theorem guides us to evaluate \( F(2) - F(-2) \). This gives the result as \( 4 - (-12) = 16 \), which is the value of the definite integral. This theorem not only simplifies calculations but also provides a robust bridge between the derivative and integral concepts.