Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{0}^{1}\left(2 x^{4}-3 x^{2}+5\right) d x $$

Step-by-Step Solution

Answer

The value of the integral is \(\frac{22}{5}\).

Step 1: Identify the Antiderivative

The first step in evaluating a definite integral using the Second Fundamental Theorem of Calculus is to find the antiderivative of the integrand. The integrand is \( 2x^4 - 3x^2 + 5 \). To find the antiderivative, integrate each term separately:- The antiderivative of \(2x^4\) is \(\frac{2}{5}x^5\).- The antiderivative of \(-3x^2\) is \(-x^3\).- The antiderivative of \(5\) is \(5x\).Combining these, the antiderivative \(F(x)\) is \(\frac{2}{5}x^5 - x^3 + 5x\).

Step 2: Apply the Fundamental Theorem of Calculus

According to the Second Fundamental Theorem of Calculus, the definite integral from \(a\) to \(b\) of \(f(x)\) is \(F(b) - F(a)\), where \(F(x)\) is the antiderivative found in the previous step. Here, \(a = 0\) and \(b = 1\).

Step 3: Evaluate the Antiderivative at the Bounds

Compute \(F(1)\):\[F(1) = \frac{2}{5}(1)^5 - (1)^3 + 5(1) = \frac{2}{5} - 1 + 5 = \frac{2}{5} - \frac{5}{5} + \frac{25}{5} = \frac{22}{5}\]Compute \(F(0)\):\[F(0) = \frac{2}{5}(0)^5 - (0)^3 + 5(0) = 0\]

Step 4: Subtract the Results

Subtract the evaluation at the lower bound from the evaluation at the upper bound:\[F(1) - F(0) = \frac{22}{5} - 0 = \frac{22}{5}\]

Key Concepts

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is a crucial concept that connects differentiation and integration. It asserts that if you have a continuous function, such as \[f(x) = 2x^4 - 3x^2 + 5,\] its antiderivative, denoted as \(F(x)\), can be used to calculate the definite integral over an interval from \(a\) to \(b\).

• This theorem simplifies the process by allowing you to evaluate the function's antiderivative at specific bounds.
• Specifically, it states that the definite integral of \(f(x)\) from \(a\) to \(b\) is \(F(b) - F(a)\).
• By finding the antiderivative and knowing your limits, you can determine the accumulated area under the curve of \(f(x)\) between \(a\) and \(b\).

This theorem is a powerful tool in calculus, particularly for solving problems involving definite integrals.

Antiderivative

Antiderivatives are at the heart of integral calculus and are essentially the reverse of derivatives. Given a function \(f(x)\), its antiderivative \(F(x)\) satisfies the property that when you differentiate \(F(x)\), you get back \(f(x)\).

• To find the antiderivative of a polynomial function, integrate each term separately. This involves increasing the exponent of each term by one and dividing by the new exponent. For example: \[\int 2x^4 \, dx = \frac{2}{5}x^5.\]
• Consider \(2x^4 - 3x^2 + 5\). Its antiderivative is: \[F(x) = \frac{2}{5}x^5 - x^3 + 5x.\]

Finding the antiderivative is a preparatory step to apply the Second Fundamental Theorem of Calculus. Proper integration techniques are essential here.

Calculus Problem Solving

Solving calculus problems involves a systematic approach that combines different rules and theorems of calculus.

• Identify the function that you need to integrate over a specific interval.
• Determine its antiderivative, understanding that each term is treated separately in polynomial functions.
• Apply the Second Fundamental Theorem of Calculus to evaluate the definite integral by substituting the interval bounds into the antiderivative.
• Subtract the lower bound evaluation from the upper bound evaluation.

This step-by-step method helps in breaking down complex integration problems into manageable parts, leading to effective solutions.

Integral Evaluation

Evaluating a definite integral involves calculating the net area under the curve represented by the integrand over a specific interval.

• Start by identifying the antiderivative, an essential process to transition from the function back to an expression that can be used for evaluation.
• For the definite integral \( \int_{0}^{1}(2x^4 - 3x^2 + 5) \, dx\), we found that the antiderivative is \( F(x) = \frac{2}{5}x^5 - x^3 + 5x \).
• Evaluate this function at the bounds: \( F(1) = \frac{22}{5} \) and \( F(0) = 0 \).
• Finally, subtract to find the evaluated integral: \( F(1) - F(0) = \frac{22}{5} \).

The process of integral evaluation results in a numerical value representing the area of concern, showcasing the utility and precision of calculus in solving real-world problems.