Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral. $$ \int_{1}^{4} \sqrt{\frac{2}{x}} d x $$

Step-by-Step Solution

Answer

The evaluated definite integral is \(4/3\).

Step 1: Rewrite the function

First, it is helpful to rewrite the function as a power of x. The function \( \sqrt{2/x} \) inside the integral can be rewritten as \( 2x^{-1/2} \) to make the next step, which is integrating, easier.

Step 2: Integrate

Next, find the antiderivative of this function. Use the power rule for integration, which states that the integral of x^n dx is \( (x^{n+1})/(n+1) + C \), where C is the constant of integration. Here, n is -1/2. So, its antiderivative is:\( \int 2x^{-1/2} dx = 2*(2/3)*x^{(1/2)}, where C=0 as we are computing for a definite integral. So the antiderivative of the function is \( (4/3)x^{1/2} \).

Step 3: Compute the definite integral

Substitute the limits 1 and 4 into the antiderivative: \( (4/3)*4^{1/2} - (4/3)*1^{1/2} \). So, the definite integral is \( (4/3)*2 - (4/3)*1 = 4/3 - 4/3 = 4/3.

Key Concepts

Antiderivative

The concept of an antiderivative is essential in calculus. It is the reverse process of differentiation, meaning if you have the derivative of a function, finding the antiderivative essentially means figuring out the original function before it was differentiated.

An antiderivative of a function 'f' is another function F, such that the derivative of F is f. It's denoted by the integral symbol without bounds, and it includes an arbitrary constant, often represented by 'C', because differentiation removes constants. In our exercise, after rewriting \( \sqrt{2/x} \) as \( 2x^{-1/2} \) to simplify the function, the antiderivative is sought to evaluate the definite integral.

Power Rule for Integration

The power rule for integration is a fundamental tool when solving integrals of algebraic functions. It states that for any real number n different from -1, if you have a function of \( x^n \) and you wish to find its integral, you increase the exponent by one and divide by the new exponent, then add the constant of integration C. In formula terms, it is \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \).

This rule greatly simplifies the process of integration. When applying the power rule to the rewritten function \( 2x^{-1/2} \) in our problem, the antiderivative calculated is \( (4/3)x^{1/2} \), following the power rule steps.

Integral Evaluation

Evaluating a definite integral involves calculating the exact area under a curve between two specified points, which are the upper and lower limits of the integral. This area represents the accumulated total of the quantified variable. After finding the antiderivative of the function within the integral, the evaluation is done by applying the Fundamental Theorem of Calculus, which involves substituting the upper limit into the antiderivative, subtracting the result of the lower limit substituted into the antiderivative.

In our exercise, after finding the antiderivative, we evaluate the definite integral from 1 to 4, which gives us \( (4/3)*4^{1/2} - (4/3)*1^{1/2} \), simplifying down to \( 4/3 - 4/3 = 0 \). It looks like there might be a miscalculation in the original problem since evaluating the integral should not result in a fraction such as \( 4/3 \), rather it should be simply \( 0 \).