Problem 20 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the integral. \(\int_{0}^{1} \frac{r^{3}}{\sqrt{4+r^{2}}} d r\)

Step-by-Step Solution

Answer

The value of the integral is approximately 7.104.

Step 1: Identify the Substitution

Notice that the integral involves an expression within a square root that suggests a substitution. Use the substitution \( u = 4 + r^2 \), such that \( du = 2r \, dr \). Hence, \( r \, dr = \frac{1}{2} du \).

Step 2: Adjust the Limits of Integration

Translate the limits of integration from \( r \) to \( u \). When \( r = 0 \), \( u = 4 + 0^2 = 4 \). When \( r = 1 \), \( u = 4 + 1^2 = 5 \). Therefore, the new limits are from \( u = 4 \) to \( u = 5 \).

Step 3: Substitute and Simplify the Integral

Substitute \( u \) into the integral. The integral becomes: \[ \int_{4}^{5} \frac{\left( \sqrt{u} - 4 \right)^2}{2\sqrt{u}} du = \frac{1}{2} \int_{4}^{5} \left( u - 8\sqrt{u} + 16 \right) \left( \frac{1}{\sqrt{u}} \right) du \] This simplifies to: \[ \frac{1}{2} \int_{4}^{5} \left( \sqrt{u} - 8 + \frac{16}{\sqrt{u}} \right) du \]

Step 4: Evaluate the Simplified Integral

Split the integral into three simpler integrals: \[ \frac{1}{2} \left[ \int_{4}^{5} \sqrt{u} \, du - 8 \int_{4}^{5} du + 16 \int_{4}^{5} \frac{1}{\sqrt{u}} \, du \right] \]Now evaluate each:1. For \( \int \sqrt{u} \, du \), integrate to get \( \frac{2}{3} u^{3/2} \).2. For \( \int du \), integrate to get \( u \).3. For \( \int \frac{1}{\sqrt{u}} \, du \), integrate to get \( 2\sqrt{u} \).

Step 5: Compute the Definite Integrals

Compute each definite integral:1. Compute \( \frac{2}{3} u^{3/2} \) from 4 to 5. The value is \( \frac{2}{3}(5^{3/2} - 4^{3/2}) \).2. Compute \( u \) from 4 to 5. The value is \( 5 - 4 = 1 \).3. Compute \( 2\sqrt{u} \) from 4 to 5. The value is \( 2(\sqrt{5} - \sqrt{4}) \).Combine these results, applying the factor of \( \frac{1}{2} \).

Step 6: Combine the Results

Substitute the computed values into the simplified expression:\[ \frac{1}{2} \left[ \frac{2}{3}(\sqrt{5}^3 - \sqrt{4}^3) - 8(1) + 16(\sqrt{5} - \sqrt{4}) \right] \]Simplify this expression to find the total. The final answer is \( \frac{2}{3} \times (5\sqrt{5} - 8) + 8 \times (\sqrt{5} - 2) \).

Step 7: Final Simplification and Answer

Simplify the complex arithmetic expression to arrive at the final answer. Use approximations to manage the exact terms or convert into decimal form to check.

Key Concepts

Integration Techniques

When evaluating the definite integral \( \int_{0}^{1} \frac{r^{3}}{\sqrt{4+r^{2}}} d r \), we employ strategic integration techniques to simplify the process. These methods help in tackling complex integrals by transforming them into more manageable forms.

Two common integration techniques include:

• Substitution Method: Replaces a complex part of the integrand with a single variable.
• Integration by Parts: Decomposes products into simpler integrals, though it's not used in this problem.

Selecting the appropriate technique can significantly simplify your calculations and lead to a successful evaluation.

Substitution Method

The substitution method is often used in integration to make the integral simpler by converting it into a function of a single variable.

In this exercise, we identify that the square root expression \( \sqrt{4 + r^2} \) suggests a substitution. By letting \( u = 4 + r^2 \) and calculating the differential as \( du = 2r \, dr \), we convert \( \frac{r^3}{\sqrt{4+r^2}} \) into an expression in terms of \( u \).

The pivotal step is to express \( r \, dr \) in terms of \( du \):
\[ r \, dr = \frac{1}{2} du.\] This allows us to rewrite the integral, making it simpler to evaluate.

Limits of Integration

While performing a substitution in definite integrals, updating the limits of integration is crucial to ensure correct evaluation.

Given the original limits \( r = 0 \) to \( r = 1 \), we substitute into our expression for \( u \):

• When \( r = 0 \), \( u = 4 + 0^2 = 4 \).
• When \( r = 1 \), \( u = 4 + 1^2 = 5 \).

Thus, the integral becomes \( \int_{4}^{5} \), preserving the definite nature and ensuring the smooth transition from the old variable \( r \) to the new variable \( u \).

Simplification of Integrals

Simplifying the integrals is about breaking down complex expressions into smaller and more easily computable parts. After performing the substitution and adjusting the limits, the integral \( \int_{4}^{5} \frac{\left( \sqrt{u} - 4 \right)^2}{2\sqrt{u}} du \) becomes more manageable.

We first expand the expression within the integral:
\[ \frac{1}{2} \int_{4}^{5} \left( u - 8\sqrt{u} + 16 \right) \left( \frac{1}{\sqrt{u}} \right) du \]
Splitting this into simpler integrals provides a set of simpler components:
\[ \frac{1}{2} \left[ \int_{4}^{5} \sqrt{u} \, du - 8 \int_{4}^{5} du + 16 \int_{4}^{5} \frac{1}{\sqrt{u}} \, du \right] \]
This approach allows us to evaluate each separately and combine the results, leading to the simplification of the problem.