Problem 1 - Advanced Engineering Mathematics

Question

$$\int_{-2}^{2} x x^{2} d x=\left.\frac{1}{4} x^{4}\right|_{-2} ^{2}=0$$

Step-by-Step Solution

Answer

The value of the integral is 0.

Step 1: Identify the Integral

We need to evaluate the definite integral of the function \( x \times x^2 \) from \(-2\) to \(2\). Simplifying, we have the integral \( \int_{-2}^{2} x^3 \, dx \).

Step 2: Find the Indefinite Integral

Next, find the antiderivative of \( x^3 \). The power rule tells us that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \). So, the antiderivative of \( x^3 \) is \( \frac{x^4}{4} \).

Step 3: Evaluate the Definite Integral

We now evaluate the definite integral using the limits of integration. Apply the limits to \( \frac{1}{4} x^4 \): \[ \left.\frac{1}{4} x^4 \right|_{-2}^{2} = \frac{1}{4}(2^4) - \frac{1}{4}((-2)^4) \]

Step 4: Simplify the Expression

Calculate \( 2^4 \) and \((-2)^4\). Both equal 16. Therefore, \[ \frac{1}{4}(16) - \frac{1}{4}(16) = 4 - 4 = 0 \]

Step 5: Conclusion

Since the values at the upper and lower limits are the same, the value of the integral is 0.

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus used to find the antiderivative of power functions. It is particularly useful when dealing with polynomial expressions.

In general, when you have an expression of the form \(x^n\), the antiderivative can be found quickly using the formula:

• Antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\).

This tells us that you increase the exponent by one and divide by the new exponent.

For example, in our exercise, we have \(x^3\). By applying the power rule, the antiderivative becomes \(\frac{x^4}{4}\). This step simplifies integrating polynomial functions and serves as a core tool in understanding more complex functions in calculus.

Antiderivative

The antiderivative is the opposite process of differentiation. While differentiation gives you the rate of change, the antiderivative provides you the original function before it was differentiated.

This is why it's also called the "indefinite integral." When taking an antiderivative, you're essentially asked to find a function that, when differentiated, results in the original function you started with. In the given exercise, the problem asks us to take the antiderivative of \(x^3\). By using the power rule as a guide, we found that the antiderivative is \(\frac{x^4}{4}\). This is crucial when calculating definite integrals, as you need to know the antiderivative to apply the limits of integration correctly.

Limits of Integration

The limits of integration provide the range over which you're calculating the definite integral. They transform the indefinite integral into a complete value, essentially finding the total accumulation over that specific interval. In our example, the limits are

• lower limit \(-2\)
• upper limit \(2\)

These limits mean we're calculating the area under the curve \(x^3\) from \(-2\) to \(2\).You apply these limits after finding the antiderivative, plugging the upper limit into the antiderivative, and then subtracting the result of plugging the lower limit. For our problem, this looks like:

• Find \(\frac{1}{4} \times 2^4\)
• Subtract \(\frac{1}{4} \times (-2)^4\)

The integral evaluates to zero because both upper and lower limits yield the same value. Hence, there's no net area, as the positive and negative areas cancel each other out.