Problem 48 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the definite integral. \(\int_{0}^{a} x \sqrt{a^{2}-x^{2}} d x\)

Step-by-Step Solution

Answer

The definite integral evaluates to \( \frac{5}{3} a^3 \).

Step 1: Recognize Trigonometric Substitution

The integral \[ \int_{0}^{a} x \sqrt{a^{2}-x^{2}} \, dx \]is well-suited for trigonometric substitution because of the radical expression \( \sqrt{a^2 - x^2} \). Since it is of the form \( \sqrt{a^2 - x^2} \), we use the substitution \( x = a \sin \theta \).

Step 2: Determine Substitutions

From the substitution \( x = a \sin \theta \), we find that \( dx = a \cos \theta \, d\theta \). The expression \( \sqrt{a^2 - x^2} \) becomes \( \sqrt{a^2 - (a \sin \theta)^2} = a \cos \theta \). Therefore, the substitution in the integral gives: \[ \int x \sqrt{a^2 - x^2} \, dx = \int a \sin \theta \cdot a \cos \theta \cdot a \cos \theta \, d\theta = a^3 \int \sin \theta \cos^2 \theta \, d\theta. \]

Step 3: Set Limits of Integration

The limits of integration change according to the substitution \( x = a \sin \theta \):- When \( x = 0 \), \( \theta = 0 \). - When \( x = a \), \( \theta = \frac{\pi}{2} \).Thus, our integral becomes:\[ a^3 \int_{0}^{\frac{\pi}{2}} \sin \theta \cos^2 \theta \, d\theta. \]

Step 4: Simplify the Integral using Trigonometric Identity

Now, we simplify the integrand using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \). This becomes:\[ a^3 \int_{0}^{\frac{\pi}{2}} \sin \theta (1 - \sin^2 \theta) \, d\theta = a^3 \int_{0}^{\frac{\pi}{2}} (\sin \theta - \sin^3 \theta) \, d\theta. \]

Step 5: Integrate Each Term Separately

We can split the integral into two parts and integrate each separately:\[ a^3 \left( \int_{0}^{\frac{\pi}{2}} \sin \theta \, d\theta - \int_{0}^{\frac{\pi}{2}} \sin^3 \theta \, d\theta \right). \]The first integral \( \int \sin \theta \, d\theta = -\cos \theta \), resulting in:\( a^3 \left[ -\cos \theta \right]_{0}^{\frac{\pi}{2}} = a^3 (0 + 1) = a^3. \)

Step 6: Evaluate Integral of \( \sin^3 \theta \)

To evaluate \( \int \sin^3 \theta \, d\theta \), use the identity \( \sin^3 \theta = \sin \theta (1 - \cos^2 \theta) \). This translates to:\[ \int \sin^3 \theta \, d\theta = \int \sin \theta - \sin \theta \cos^2 \theta \, d\theta. \]The first term \( \int \sin \theta \, d\theta = -\cos \theta \), and for \( \int \sin \theta \cos^2 \theta \, d\theta \), use substitution \( u = \cos \theta \), \( du = -\sin \theta \, d\theta \).

Step 7: Continue with Integration by Substitution

Substitute \( \int \sin \theta \cos^2 \theta \, d\theta = \int \cos^2 \theta \, (-du) \), which becomes \[ \int (1-u^2) \, du = u - \frac{u^3}{3}. \]Back-substitute \( u = \cos \theta \) in the result, obtaining:\[ \left[ \cos \theta - \frac{\cos^3 \theta}{3} \right]_{0}^{\frac{\pi}{2}} = \left( 0 - \frac{0}{3} \right) - \left( 1 - \frac{1}{3} \right) = -\frac{2}{3}. \]

Step 8: Combine and Conclude the Integral Evaluation

Substituting back and multiplying by \( a^3 \):\[ a^3 \left( 1 + \frac{2}{3} \right) = \frac{5}{3} a^3. \]This is the value of the definite integral \( \int_{0}^{a} x \sqrt{a^{2} - x^{2}} \, dx \).

Key Concepts

Trigonometric Substitution

When solving definite integrals with radical expressions like \( \sqrt{a^2 - x^2} \), trigonometric substitution can simplify the process. This is because trigonometric functions naturally express geometric relationships encapsulated by the Pythagorean identity. By substituting \( x = a \sin \theta \), we leverage the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to transform the radical into a more manageable form. After substitution, the expression \( \sqrt{a^2 - x^2} \) simplifies to \( a \cos \theta \). This transformation is crucial for reducing complex integrals into simpler trigonometric forms, making them easier to handle and evaluate.

Limits of Integration

Integrating with trigonometric substitution affects the limits of integration. Originally, our limits are in terms of \( x \), running from 0 to \( a \). When substituting \( x = a \sin \theta \), we need to adjust these limits to correspond to \( \theta \).

• When \( x = 0 \), solving \( 0 = a \sin \theta \) gives us \( \theta = 0 \).
• When \( x = a \), solving \( a = a \sin \theta \) results in \( \theta = \frac{\pi}{2} \).

Thus, the new limits of integration are \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \). Adjusting the integration limits ensures the integral remains accurate and corresponds exactly to the domain of the transformed integrand.

Trigonometric Identity

Trigonometric identities are powerful tools for simplifying integrals. Specifically, in our integral, the identity \( \cos^2 \theta = 1 - \sin^2 \theta \) helps break down \( \cos^2 \theta \) into simpler components, aiding in the integration process. Using this identity, the integral \( a^3 \int_{0}^{\frac{\pi}{2}} \sin \theta \cos^2 \theta \, d\theta \) transforms into \( a^3 \int_{0}^{\frac{\pi}{2}} (\sin \theta - \sin^3 \theta) \, d\theta \).
These identities transform complex products of sines and cosines into functions that are easier to integrate separately, thus making the evaluation process more tractable.

Integration by Substitution

Integration by substitution is a technique used to evaluate integrals by reversing the chain rule for differentiation. In the process of solving our integral, after some trigonometric reductions, we encounter terms that require further manipulation. For example, when dealing with \( \int \sin \theta \cos^2 \theta \, d\theta \), we make a secondary substitution \( u = \cos \theta \), resulting in \( du = -\sin \theta \, d\theta \).
This substitution modifies the integral into \( \int (1-u^2) \, du \), which is more straightforward to evaluate. Integration by substitution is essentially about finding an equivalent integral with a simpler form, making it easier to compute the antiderivative and evaluate the overall integral. By substituting back, we finalize our solution, maintaining the continuity and accuracy of our integral evaluation.