Problem 36 - Calculus

Question

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

\( \frac{2}{3} \)

Step 1: Identify Substitution

The integrand contains the expression \( x^3 + 1 \). To simplify, set \( u = x^3 + 1 \). Then the differential \( du \) is derived as \( du = 3x^2 \, dx \). Notice that \( 3x^2 \, dx \) is present in the integrand, which matches perfectly with \( du \).

Step 2: Change of Limits

Since \( u = x^3 + 1 \), compute the new limits for \( u \). When \( x = -1 \), \( u = (-1)^3 + 1 = 0 \). When \( x = 0 \), \( u = 0^3 + 1 = 1 \). The limits of integration change from \( x = -1 \) to \( x = 0 \) into \( u = 0 \) to \( u = 1 \).

Step 3: Rewrite the Integral

With the substitution, the integral \( \int_{-1}^{0} \sqrt{x^3+1} \cdot 3x^2 \, dx \) becomes \( \int_{0}^{1} \sqrt{u} \, du \). This simplifies the expression significantly, focusing on the square root function.

Step 4: Integrate with New Limits

Integrate \( \int_{0}^{1} \sqrt{u} \, du \). The antiderivative of \( \sqrt{u} = u^{1/2} \) is \( \frac{2}{3}u^{3/2} \). Evaluate this from \( u = 0 \) to \( u = 1 \).

Step 5: Evaluate the Definite Integral

Substitute the limits into the antiderivative. Compute \( \frac{2}{3}(1)^{3/2} - \frac{2}{3}(0)^{3/2} = \frac{2}{3} - 0 = \frac{2}{3} \).

Key Concepts

Substitution Rule

When dealing with definite integrals, one of the most powerful techniques is the substitution rule. This method simplifies complicated integrals by transforming the variable of integration. In the given exercise, we are tasked with evaluating the integral \( \int_{-1}^{0} \sqrt{x^{3}+1}\left(3 x^{2}\right) d x \). To employ the substitution rule, we look for a part of the integrand that can be substituted to make the integral easier to handle. Here, the expression \( x^3 + 1 \) stands out as a suitable candidate for substitution.

• Set \( u = x^3 + 1 \).
• Calculate the differential: \( du = 3x^2 \, dx \). Notice that the integrand actually contains \( 3x^2 \, dx \), allowing us to directly replace it with \( du \).

This substitution transforms the integral from a complex expression into a much simpler one. You see, the challenge of dealing with the original variable, \( x \), vanishes. The problem becomes solving an integral with respect to \( u \).
The substitution rule works similarly to reversing the chain rule in differentiation, where you peel back the layers of complexity by changing the focus to a different variable.

Limits of Integration

When initially given a definite integral, you have specific limits based on the variable of integration. However, after making a substitution, these limits must be adjusted to correspond with the new variable. This is crucial for properly evaluating the transformed integral.
In this exercise:

• The original limits are \( x = -1 \) to \( x = 0 \).
• Upon substitution \( u = x^3 + 1 \), new limits are needed based on \( u \).
• When \( x = -1 \), substituting into \( u = x^3 + 1 \) gives \( u = 0 \).
• When \( x = 0 \), \( u = 1 \).

Therefore, the limits of integration change to \( u = 0 \) and \( u = 1 \). By adjusting these limits, you ensure the new integral accurately reflects the desired interval. This step highlights an important part of using substitution in definite integrals—keeping track of your endpoints during transformation.

Antiderivative Evaluation

Once you have completed the substitution process, the next step involves finding the antiderivative of the transformed integral. The revised problem in this exercise after substitution became \( \int_{0}^{1} \sqrt{u} \, du \).
Let's break it down:

• The function \( \sqrt{u} = u^{1/2} \), requires you to find its antiderivative.
• The antiderivative of \( u^{1/2} \) is \( \frac{2}{3}u^{3/2} \).

Once you derive the antiderivative, use the Fundamental Theorem of Calculus to evaluate the integral with the new limits. Here we substitute for \( u \):

• Plugging in \( u = 1 \) gives \( \frac{2}{3}(1)^{3/2} = \frac{2}{3} \).
• Plugging in \( u = 0 \) gives \( \frac{2}{3}(0)^{3/2} = 0 \).

Subtract the results to find the value of the definite integral, which is \( \frac{2}{3} - 0 = \frac{2}{3} \).
This final step, known as antiderivative evaluation, completes the process. By working through these steps, you convert a complex integral into something manageable and find the area under the curve as specified by the limits of integration.