Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the definite integral. $$\int_{-1}^{3} 2 x^{2} d x$$

Step-by-Step Solution

Answer

The value of the definite integral \(\int_{-1}^{3} 2 x^{2} d x\) is \(\frac{56}{3}\).

Step 1: Find the antiderivative of the function

To find the antiderivative of the function, we will use the power rule for integration, which states that $$\int x^n d x = \frac{1}{n + 1}x^{n+1} + C$$ where `n` is a constant and `C` is the integration constant. Applying the power rule to the function, we get: $$\int 2x^2 d x = 2\int x^2 d x = 2\left(\frac{1}{2+1}x^{2+1}\right) + C = \frac{2}{3}x^3 + C$$ The antiderivative of the function is \(\frac{2}{3}x^3 + C\).

Step 2: Apply the Fundamental Theorem of Calculus

Now that we have the antiderivative of the function, we can apply the Fundamental Theorem of Calculus. We have: $$F(x) = \frac{2}{3}x^3 + C$$ We can now find the definite integral over the interval [-1, 3]: $$\int_{-1}^{3} 2 x^{2} d x = F(3) - F(-1)$$ Now, we just substitute the values in the antiderivative function: $$F(3) = \frac{2}{3}(3)^3 = \frac{2}{3}(27) = 18$$ $$F(-1) = \frac{2}{3}(-1)^3 = \frac{2}{3}(-1) = -\frac{2}{3}$$ Thus, the definite integral is: $$\int_{-1}^{3} 2 x^{2} d x = F(3) - F(-1) = 18 - (-\frac{2}{3})$$ $$\int_{-1}^{3} 2 x^{2} d x = 18 + \frac{2}{3} = \frac{54}{3} + \frac{2}{3} = \frac{56}{3}$$ The value of the definite integral is \(\frac{56}{3}\).

Key Concepts

Antiderivative

The concept of an antiderivative is crucial in calculus. It refers to a function whose derivative is the original function given in the integral. When you take the derivative of the antiderivative, you end up with the original function inside the integral. For instance, if you're given a function such as \( 2x^2 \), finding an antiderivative involves reversing the differentiation process. In the context of integration, finding an antiderivative is necessary to evaluate definite and indefinite integrals. An important note is the addition of the constant \( C \). When finding an indefinite integral or antiderivative, this "constant of integration" is included because differentiating multiple functions could yield the same derivative; the difference could just be a constant term that disappears in differentiation.

Power Rule for Integration

The Power Rule for Integration simplifies finding antiderivatives of power functions. It is especially useful for functions like \( x^n \) where \( n \) is a constant. The rule states: \[ \int x^n \, dx = \frac{1}{n + 1}x^{n+1} + C \] This formula allows you to increase the exponent by 1 and divide by the new exponent. Let's consider our function \( 2x^2 \). Using the power rule, we treat it as \( x^2 \) first:

• Raise the exponent from 2 to 3.
• Divide by the new exponent 3.

So, for \( x^2 \), the antiderivative results in \( \frac{1}{3}x^3 + C \). Multiplying by 2 (from our original function), we get \( \frac{2}{3}x^3 + C \). Hence, the Power Rule efficiently helps us handle polynomials and other power functions in integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, two main concepts of calculus, in a powerful way. It essentially provides a method to evaluate definite integrals using antiderivatives. The theorem states that if \( F \) is an antiderivative of \( f \) on an interval \( [a, b] \), then:\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]In simpler terms, it means you find the antiderivative of the function, evaluate it at the upper limit of the integral, and then subtract the value of the antiderivative evaluated at the lower limit.In our given exercise, after finding the antiderivative \( F(x) = \frac{2}{3}x^3 + C \), we evaluate it at \( x = 3 \) and \( x = -1 \):

• Calculate \( F(3) = 18 \)
• Calculate \( F(-1) = -\frac{2}{3} \)
• Find the difference: \( F(3) - F(-1) \)

This process efficiently provides the value of the definite integral, showing the power of the Fundamental Theorem in bridging concepts of calculus.