Problem 1 - Calculus

Question

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{0}^{2} x^{3} d x $$

Step-by-Step Solution

Answer

The value of the definite integral is 4.

Step 1: Identify the Function

The given definite integral is \( \int_{0}^{2} x^{3} \; dx \). Here, the integrand is \( f(x) = x^3 \).

Step 2: Find the Antiderivative

To apply the Second Fundamental Theorem of Calculus, we need the antiderivative of \( f(x) = x^3 \). The antiderivative of \( x^3 \) is given by \( F(x) = \frac{x^4}{4} + C \), where \( C \) is the constant of integration.

Step 3: Apply the Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then \( \int_{a}^{b} f(x) \; dx = F(b) - F(a) \). For this problem, \( a = 0 \) and \( b = 2 \). Thus, we calculate \( F(2) - F(0) \).

Step 4: Evaluate the Antiderivative at the Bounds

Substitute the upper limit 2 into the antiderivative: \( F(2) = \frac{2^4}{4} = \frac{16}{4} = 4 \). Substitute the lower limit 0 into the antiderivative: \( F(0) = \frac{0^4}{4} = 0 \).

Step 5: Calculate the Definite Integral

Finally, using the values calculated, find \( F(2) - F(0) \): \( 4 - 0 = 4 \).

Key Concepts

Definite Integral Evaluation

Evaluating a definite integral means finding the accumulation of areas under a curve, from one point to another. In this exercise, the integral \( \int_{0}^{2} x^{3} \, dx \) represents the area under the curve of the function \( x^3 \) between the limits 0 and 2. This is accomplished through the Second Fundamental Theorem of Calculus.

To evaluate a definite integral, follow these general steps:

• Determine the function you are integrating, also known as the integrand.
• Find the antiderivative of the function.
• Evaluate this antiderivative at the upper and lower limits and then subtract the values.

These steps help transform a function's geometrical aspects, or shapes under the curve, into numerical values. Thus, you can obtain the total area exact from the calculated result.

Antiderivative

An antiderivative is the reverse process of differentiation, often related to the term "indefinite integral." Finding an antiderivative involves determining a function that, when differentiated, results in the original function, also known as the integrand.

For the function \( f(x) = x^3 \), its antiderivative is \( F(x) = \frac{x^4}{4} + C \), where \( C \) is a constant. This constant appears because the derivative of a constant is zero, thus disappearing when derived.

To solve a definite integral as we do here, the constant \( C \) is not required because it's subtracted out when evaluating the limits of integration. Therefore, the antiderivative takes a primary role in determining areas under the curve within the specified limits.

Calculus Step-by-Step Solution

Solving a calculus problem like this one involves a systematic approach, ensuring each step leads logically to the next. Here's how we tackle this specific problem:

**Step 1**: Recognize the Function - Here, the function or integrand is \( f(x) = x^3 \).
**Step 2**: Find the Antiderivative - Consider \( F(x) = \frac{x^4}{4} + C \). Ignore the constant \( C \) for definite integrals.
**Step 3**: Apply the Second Fundamental Theorem of Calculus - This theorem is central to bridging antiderivatives with definite integrals. It connects the values of \( F \) over the bounds \([0, 2]\).

• \( F(b) = F(2) = \frac{2^4}{4} = 4 \)
• \( F(a) = F(0) = \frac{0^4}{4} = 0 \)
• Final Calculation: \( F(b) - F(a) = 4 - 0 = 4 \)

This straightforward solution maps each integral component to its outcome, making it easier to comprehend how calculus tools guide us from mathematical expressions to calculated results.