Problem 58 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{27} \sqrt{3 x} d x $$

Step-by-Step Solution

Answer

The value of the integral is 162.

Step 1: Understand the Problem

We need to evaluate the definite integral \( \int_{0}^{27} \sqrt{3x} \, dx \). This involves finding the antiderivative of the function \( \sqrt{3x} \) and then calculating the definite integral from 0 to 27.

Step 2: Rewrite the Integrand as a Power Function

The function \( \sqrt{3x} \) can be rewritten using exponents as \( (3x)^{1/2} \). This will allow us to use the power rule for integration.

Step 3: Apply the Power Rule for Integration

The antiderivative of \( (3x)^{1/2} \) can be found by using the formula for the integral of \( x^n \), which is \( \frac{x^{n+1}}{n+1} \). Here, let \( u = 3x \) then \( du = 3 \, dx \), and the integral turns into: \[ \int (3x)^{1/2} \, dx = \int u^{1/2} \frac{du}{3} \].

Step 4: Solve the Antiderivative using Substitution

Taking \( u = 3x \), the integral \( \int u^{1/2} \frac{du}{3} \) becomes \( \frac{1}{3} \int u^{1/2} \, du \). The integral of \( u^{1/2} \) is \( \frac{2}{3} u^{3/2} \), thus the antiderivative in terms of \( u \) becomes \( \frac{2}{9} u^{3/2} + C \). Re-substitute \( u = 3x \) to get: \( \frac{2}{9} (3x)^{3/2} + C \).

Step 5: Evaluate the Definite Integral

Now that we have the antiderivative, evaluate from 0 to 27: \[ \left[ \frac{2}{9} (3x)^{3/2} \right]_0^{27} = \frac{2}{9} ((3 \times 27)^{3/2} - (3 \times 0)^{3/2}) = \frac{2}{9} (81^{3/2} - 0) \]. Simplify \( 81^{3/2} \) to find that it equals 729.

Step 6: Final Calculation

Calculate \( \frac{2}{9} \times 729 \), which simplifies to 162. Therefore, the value of the integral is 162.

Key Concepts

Antiderivative

The antiderivative is an essential concept in calculus. It refers to the reverse process of differentiation. While differentiation gives the rate of change or the slope of a curve, finding an antiderivative means determining a function whose derivative will return to the original function. When you find an antiderivative, you add a constant, often denoted as "C," because the derivative of a constant is zero. This represents an entire family of functions with similar slopes but different vertical positions.

For definite integrals, however, this constant "C" gets cancelled when evaluating the upper and lower bounds. In this specific exercise, we focus on finding the antiderivative of the function \( \sqrt{3x} \), which is a crucial step before we calculate the definite integral across specified limits. By rewriting \( \sqrt{3x} \) as \( (3x)^{1/2} \), we prepare it for easier integration using the power rule.

Power Rule for Integration

The power rule for integration is a fundamental tool for finding antiderivatives of power functions. It states that for a function \( x^n \), the integral is \( \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). This rule helps simplify and solve integrals efficiently.

In the problem we are tackling, we first transform the function \( \sqrt{3x} \) to \( (3x)^{1/2} \). Applying the power rule here involves incrementing the exponent by 1, yielding \( \frac{1}{2} + 1 = \frac{3}{2} \). Then we divide by the new exponent \( \frac{3}{2} \), followed by dividing the outside expression by 3 due to the substitution of \( u = 3x \). This method allows us to seamlessly integrate functions that may initially seem complex, such as roots and polynomial expressions.

Substitution Method

The substitution method, often referred to as "u-substitution," is a technique used in integration to simplify complex expressions by substituting part of the integral with a new variable. This technique is similar to the chain rule used in differentiation and is especially useful when the integral involves a composition of functions. The goal is to make the integral easier to solve by transforming it into a simpler form.

In this exercise, by letting \( u = 3x \), we derive that \( du = 3 \, dx \), allowing us to rewrite the integral \( \int (3x)^{1/2} \, dx \) as \( \frac{1}{3} \int u^{1/2} \, du \). After integration, we substitute back \( 3x \) for \( u \) to express the antiderivative in terms of the original variable. The substitution not only simplifies the calculation process but also effectively handles integrals involving non-linear transformations, making it an invaluable method in tackling a range of calculus problems."