Problem 94 - Essential Calculus

Question

Rewrite the improper integral as a proper integral using the given \(u\) -substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\cos x}{\sqrt{1-x}} d x, u=\sqrt{1-x} $$

Step-by-Step Solution

Answer

The answer will depend on the exact approximation from the Trapezoidal Rule computation for \(n=5\). This is a numerical method and it doesn't provide a symbolic result.

Step 1: Variable Substitution

We will start by substituting the \(u\) variable into the integral. Given that \(u = \sqrt{1-x}\), we can find the substitutions for \(x\) and \(d x\). Hence, \(x = 1 - u^2\) and \(d x = -2u du\). Now we substitute these expressions into the integral and also rewrite the limits of the integral from \(x=0\) to \(1\) as \(u=1\) to \(0\): \[ \int_{1}^{0} \frac{\cos (1-u^2)}{u} * -2u \, du \]

Step 2: Simplification

Before implementing the Trapezoidal Rule, simplify the integral. As you can see, the \(u\) in the denominator and the multiplied '-2u' cancel out, allowing the integral to be rewritten as: \[ -2\int_{1}^{0} \cos(1-u^2) \, du \]

Step 3: Numerical Approximation with the Trapezoidal Rule

Now, use the Trapezoidal Rule with \(n=5\) to approximate the integral. It is necessary to calculate the sub-intervals and plug in the values into the formula: \[ T_n = \frac{-2}{2*5} * \left[f(1) + 2* \left(\sum_{i=1}^{4} f(x_i) \right) + f(0) \right] \] Here \(f(x)\) represents the integrand without the integration (dx) part, and the values \(x_i\) represent the values in the sub-interval. The goal now is to compute the integral using these terms.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. It's particularly useful when an integral cannot be evaluated exactly, or when an antiderivative is difficult to find.Here's how it works:

• Divide the interval of integration into smaller sub-intervals of equal length.
• Approximate the area under the curve for each sub-interval by a trapezoid rather than a curve.
• Calculate the total area of these trapezoids to estimate the integral.

For our exercise, the integral is approximated using the Trapezoidal Rule with 5 sub-intervals (n = 5). By breaking it down into smaller parts, you simply apply the formula:\[ T_n = \frac{b-a}{2n} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] \]This calculates the sum of the first and last functional values, doubles the sum of all intermediary values, and finally adjusts by the width of each sub-interval.

u-substitution

The concept of u-substitution is a powerful tool in integral calculus, often used to simplify the integration of complex expressions. The core idea is to transform a hard-to-manage integral into a simpler one by substituting a part of the original integrand with a new variable, usually denoted as "u".In this specific exercise:

• We set \( u = \sqrt{1-x} \), aiming to simplify the original expression \( \int_{0}^{1} \frac{\cos x}{\sqrt{1-x}} \, dx \).
• The substitution involves converting \( dx \) to \( -2u \, du \) and changing \( x \) to \( 1-u^2 \).
• The integration limits shift due to substitution: when \( x = 0 \), \( u = 1 \), and when \( x = 1 \), \( u = 0 \).

The result is a more manageable integral: \( -2\int_{1}^{0} \cos(1-u^2) \, du \), demonstrating how u-substitution can greatly simplify complex integrals.

Improper Integrals

Improper integrals are those where either the interval of integration is infinite or the integrand becomes infinite within the integration limits. These types of integrals often appear in calculus when dealing with unbounded areas under curves.In this problem, the integral is improper due to the integrand \( \frac{\cos x}{\sqrt{1-x}} \) which becomes undefined at one of the endpoints (\( x = 1 \)). Here's how to handle improper integrals:

• Rewrite improper integrals as limits, where necessary.
• Consider transformations (like the u-substitution) that might transform them into proper integrals.
• Apply the appropriate numerical or analytical methods to evaluate them.

By using u-substitution, the improper nature of the integral was adjusted, allowing the numerical solution using the trapezoidal approach.