Problem 40 - Calculus Single Variable

Question

Each of the integrands involves an expression of the form \(a^{2}-b^{2} x^{2}, a^{2}+b^{2} x^{2},\) or \(b^{2} x^{2}-a^{2} .\) Use an indirect substitution of the form \(x=(a / b) \sin (\theta), x=(a / b)\) \(\tan (\theta),\) or \(x=(a / b) \sec (\theta)\) to calculate the given integral. $$ \int_{0}^{1} \sqrt{25-16 x^{2}} d x $$

Step-by-Step Solution

Answer

The integral is \(\frac{25\pi}{48} + \frac{25\sqrt{3}}{32}.\)

Step 1: Identify the Suitable Substitution

The integrand contains the expression \(25 - 16x^2\). This expression fits the form \(a^2 - b^2x^2\), where \(a = 5\) and \(b = 4\). The suitable substitution is \(x = \frac{5}{4} \sin(\theta)\), because \(\sin^2(\theta) + \cos^2(\theta) = 1\) will simplify under the square root.

Step 2: Differentiate and Change Limits of Integration

The substitution \(x = \frac{5}{4} \sin(\theta)\) implies \(dx = \frac{5}{4} \cos(\theta) d\theta\). The limits also change: when \(x = 0\), \(\theta = 0\); and when \(x = 1\), \(\theta = \frac{\pi}{6}\), since \(\sin^{-1}(\frac{4}{5}) = \frac{\pi}{6}\).

Step 3: Substitute and Simplify the Integral

Substituting \(x\) and \(dx\) in the integral gives: \[ \int_{0}^{\frac{\pi}{6}} \sqrt{25 - 16\left(\frac{5}{4} \sin(\theta)\right)^2} \cdot \frac{5}{4} \cos(\theta) d\theta. \] Simplify the expression under the square root: \[ 25 - \frac{400}{16} \sin^2(\theta) = 25 - 25\sin^2(\theta) = 25(1 - \sin^2(\theta)) = 25\cos^2(\theta). \] The expression becomes \(\sqrt{25\cos^2(\theta)} = 5\cos(\theta)\).

Step 4: Evaluate the Simplified Integral

The integral thus becomes: \[ \int_{0}^{\frac{\pi}{6}} 5\cos(\theta) \cdot \frac{5}{4} \cos(\theta) d\theta = \int_{0}^{\frac{\pi}{6}} \frac{25}{4} \cos^2(\theta) d\theta. \] To integrate \(\cos^2(\theta)\), use the identity \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\): \[ \int \frac{25}{4} \frac{1 + \cos(2\theta)}{2} d\theta = \frac{25}{8} \int (1 + \cos(2\theta)) d\theta. \] This integral is straightforward: \[ \frac{25}{8} \left( \theta + \frac{1}{2}\sin(2\theta) \right) \Bigg|_{0}^{\frac{\pi}{6}}. \]

Step 5: Calculate the Definite Integral

Substitute the limits: \[ \frac{25}{8} \left( \frac{\pi}{6} + \frac{1}{2} \sin\left( \frac{\pi}{3} \right) - \left( 0 + \frac{1}{2} \sin(0) \right) \right) = \frac{25}{8} \left( \frac{\pi}{6} + \frac{1}{2} \cdot \frac{\sqrt{3}}{2} \right). \] Simplify: \[ \frac{25}{8} \left( \frac{\pi}{6} + \frac{\sqrt{3}}{4} \right) = \frac{25}{8} \left( \frac{\pi}{6} + \frac{\sqrt{3}}{4} \right), \] giving the final result as \(\frac{25\pi}{48} + \frac{25\sqrt{3}}{32}.\)

Key Concepts

Integration

Integration is a key technique in calculus. It involves finding the integral of a function, which can be thought of as the reverse process of differentiation. When you integrate a function, you find another function whose derivative is the original function. This process is useful for calculating areas under curves, among other applications.

In our exercise, we're dealing with an integral that involves a square root expression. Integration can be approached in different ways depending on the form of the function. In this particular problem, we use a technique called trigonometric substitution to simplify the integral before solving it.

Trigonometric substitution is a method used when dealing with integrals containing expressions like \(a^2 - x^2\), \(a^2 + x^2\), or \(x^2 - a^2\). These forms are challenging to integrate directly, so they are instead rewritten using trigonometric identities to make the integration more manageable.

Definite Integral

A definite integral is an integral with specified upper and lower limits. It provides an exact numerical value, unlike an indefinite integral that results in a family of functions.

The definite integral \( \int_{a}^{b} f(x) \, dx \) is evaluated over the interval from \(a\) to \(b\). It calculates the net area between the function \(f(x)\) and the x-axis, accounting for areas above the x-axis as positive and below as negative.

For our problem, we're evaluating the definite integral \( \int_{0}^{1} \sqrt{25-16x^2} \, dx \). The limits of integration 0 to 1 mean we're interested in the area under the curve of the function from \(x=0\) to \(x=1\). This requires changing the limits to suit the transformed variable \(\theta\), which occurs through our substitution.

Indirect Substitution

Indirect substitution, especially through trigonometric substitution, is a technique that simplifies complex integrals. This method involves substituting a trigonometric expression for a variable, transforming the integral into a more manageable form.

In our example, we use the substitution \(x = \frac{5}{4} \sin(\theta)\). The idea is to use the trigonometric identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) to simplify the square root within the integral. As we substitute, the derivative \(dx = \frac{5}{4} \cos(\theta) d\theta\) is used to transform the integrand; this also necessitates changing the limits of integration based on the new substitution.

By performing these substitutions, the integral is simplified, often allowing for easier calculation, as seen when the original square root expression is reduced to a simpler form.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for any value of the occurring variables where both sides of the equality are defined. They are fundamental tools in simplifying expressions and solving integrals.

In our problem, the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) plays a crucial role. When the expression \(25 - 16x^2\) is rewritten in terms of trigonometric functions, this identity helps reduce the complexity of the integral expression. Another useful identity for our integration is \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\), which helps in integrating \(\cos^2(\theta)\).

These identities transform the integral into a more straightforward form, making the calculation of definite integrals with trigonometric substitutions successful and efficient.