Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-1}^{0} \sqrt{x^{3}+1}\left(3 x^{2}\right) d x\)

Step-by-Step Solution

Answer

The value of the integral is \( \frac{2}{3} \).

Step 1: Identify the Substitution

To solve the definite integral \( \int_{-1}^{0} \sqrt{x^{3} + 1} (3x^{2}) \, dx \), let's first identify the substitution. Notice that if \( u = x^3 + 1 \), then \( du = 3x^2 \, dx \). This suggests a perfect substitution since we have a \( 3x^2 \, dx \) factor in the integral.

Step 2: Change the Bounds

With the substitution \( u = x^3 + 1 \), we must also change the bounds of the integral. When \( x = -1 \), \( u = (-1)^3 + 1 = 0 \). When \( x = 0 \), \( u = 0^3 + 1 = 1 \). Therefore, the new bounds for \( u \) are from 0 to 1.

Step 3: Rewrite the Integral in Terms of u

Substitute \( u \) and \( du \) into the integral. The integral becomes \( \int_{0}^{1} \sqrt{u} \, du \).

Step 4: Evaluate the Integral

Evaluate the integral \( \int_{0}^{1} \sqrt{u} \, du \). This integral is equal to \( \int_{0}^{1} u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is \( \frac{2}{3}u^{3/2} \).

Step 5: Apply the Fundamental Theorem of Calculus

Using the evaluated antiderivative, apply the Fundamental Theorem of Calculus: \( \left[ \frac{2}{3}u^{3/2} \right]_{0}^{1} = \frac{2}{3}(1^{3/2}) - \frac{2}{3}(0^{3/2}) \).

Step 6: Simplify the Result

Simplify the expression from the previous step: \( \frac{2}{3} \cdot 1 - \frac{2}{3} \cdot 0 = \frac{2}{3} \). Thus, the value of the integral is \( \frac{2}{3} \).

Key Concepts

Substitution Rule

The Substitution Rule is a key technique in evaluating definite integrals, especially when direct integration is complex or not straightforward. Here's how it works: when you have an integral of a composite function, substitution can simplify the process by letting you work with a simpler form.

• Start by identifying a substitution variable, usually denoted by \( u \), which simplifies the integrand (or the function being integrated).
• In many cases, make your substitution such that \( du \), the differential of \( u \), perfectly replaces a component of the integrand.

For the integral \( \int_{-1}^{0} \sqrt{x^{3} + 1} (3x^{2}) \, dx \), notice that setting \( u = x^3 + 1 \) makes \( du = 3x^2 \, dx \). This substitution transforms the original integral into a simpler form that can then be evaluated more easily.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the process of differentiation and integration, providing a powerful method to find the value of a definite integral. Here's a simplified explanation:

• This theorem has two main parts. The first part links the concept of the antiderivative with the function's area under the curve.
• The second part tells us that if \( F \) is an antiderivative of \( f \) over an interval \([a, b]\), the definite integral of \( f \) over \([a, b]\) is \( F(b) - F(a) \).

In our step-by-step solution, after changing the integral to involve \( u \), we used the antiderivative of \( \sqrt{u} \) as \( \frac{2}{3} u^{3/2} \), then applied this theorem to compute the definite integral between the new bounds \( u = 0 \) and \( u = 1 \).

Integral Bounds

When dealing with definite integrals, it's crucial to change the limits of integration after performing a substitution. This process ensures that the bounds align with the new variable of integration.

• Originally, the integral had limits from \( x = -1 \) to \( x = 0 \).
• With the substitution \( u = x^3 + 1 \), the limits of the integral also change to \( u = 0 \) when \( x = -1 \) and \( u = 1 \) when \( x = 0 \).

This step is essential for evaluating the definite integral correctly, ensuring that all components of the integral reflect the new variable's context.

Integral Evaluation

Once substitution has been made and the integral bounds adjusted, the final step is the evaluation of the integral itself. This involves integrating the updated function over the new variable.

• Convert the integral into terms of the new variable and its bounds (as we did with \( \int_{0}^{1} \sqrt{u} \, du \)).
• Determine the antiderivative of the new integrand, in this case, \( \frac{2}{3} u^{3/2} \).
• Apply the Fundamental Theorem of Calculus to evaluate between the bounds, resulting in \( \frac{2}{3}(1^{3/2}) - \frac{2}{3}(0^{3/2}) = \frac{2}{3} \).

This process provides the final value of the definite integral, using all preceding steps to ensure accurate results.