Problem 23 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int_{1}^{\sqrt{2}} \frac{s^{2}+\sqrt{s}}{s^{2}} d s$$

Step-by-Step Solution

Answer

The value of the integral is 1.

Step 1: Rewrite the Integrand

First, simplify the expression inside the integral. The integrand \(\frac{s^{2} + \sqrt{s}}{s^{2}}\) can be split into two separate fractions: \(\frac{s^{2}}{s^{2}} + \frac{\sqrt{s}}{s^{2}}\). This further simplifies to \(1 + \frac{s^{1/2}}{s^{2}}\), which is equivalent to \(1 + s^{-3/2}\).

Step 2: Integral of Simplified Expression

Now that the integrand is simplified, evaluate the integral: \[\int_{1}^{\sqrt{2}} \left( 1 + s^{-3/2} \right) \, ds.\] You can integrate each term separately: \(\int 1 \, ds = s\) and \(\int s^{-3/2} \, ds\).

Step 3: Integrate Each Term

The integral of \(s^{-3/2}\) is \(\frac{2}{-1/2} s^{-1/2} = -2s^{-1/2}\). Therefore, the integral becomes \[\int_{1}^{\sqrt{2}} \left( 1 + s^{-3/2} \right) \, ds = \left[s - 2s^{-1/2}\right]_{1}^{\sqrt{2}}.\]

Step 4: Evaluate the Definite Integral

Substitute the upper and lower limits into \(s - 2s^{-1/2}\): \[\left[ s - 2s^{-1/2} \right]_{1}^{\sqrt{2}} = \left(\sqrt{2} - 2(\sqrt{2})^{-1/2}\right) - \left(1 - 2\cdot1^{-1/2}\right).\] Simplifying this gives \(\sqrt{2} - \sqrt{2} - 1 + 2 = 1.\)

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that represents the area under a curve, within a specific interval. This means we're summing up the values of a function to obtain a total over a defined range.
In our exercise, the definite integral is expressed as \[ \int_{1}^{\sqrt{2}} \frac{s^{2} + \sqrt{s}}{s^{2}} ds \]
This tells us to evaluate the integral from 1 to \(\sqrt{2}\).
To "evaluate" means we substitute these bounds into the antiderivative we find, then calculate the resulting difference.

• Boundaries matter! They help us understand the geometric properties of the function we're analyzing.
• The result of evaluating the definite integral provides us a numerical value, not just a function.
• This value can represent quantities such as area, displacement, or total accumulation, depending on the context of the function.

This process not only calculates the total area but also involves simplifying the function for ease of calculation.

Integral Simplification

Before tackling integrals, it's often necessary to simplify the function. This makes the integration process much more manageable. Simplifications can involve algebraic manipulation, such as breaking down complex fractions into simpler components.
For our problem:Given \[ \frac{s^{2} + \sqrt{s}}{s^{2}} \]
We performed the following steps:

• Split the fraction: \( \frac{s^{2}}{s^{2}} + \frac{\sqrt{s}}{s^{2}} \).
• Each term simplifies: \(1\) and \(s^{-3/2}\).

Now, it is much simpler to integrate term by term.
Simplifying is crucial because it transforms a complicated expression into a more straightforward form, easing overall computation while considering possible algebraic identities or transformations that can be applied. Getting to a form like \((1 + s^{-3/2})\) also aids in applying integration rules effectively.

Power Rule for Integration

When integrating, a fundamental rule we often apply is the Power Rule. This rule simplifies the integration of monomials and is essential for handling many everyday functions. The Power Rule states that for any real number \(neq -1\):\[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]
In our exercise, we use this rule to solve:

• For \(\int 1 \, ds = s\)
• For \(\int s^{-3/2} \, ds\)
• Applying the rule, it becomes \(-2s^{-1/2}\).

This method helps integrate powers of variables efficiently by increasing the power by one and then dividing by that new power.
Remember, ensuring that terms fit the Power Rule before applying it involves looking out for suitable powers and converting fractions or roots into exponent notation.