Problem 5 - Thomas Calculus

Question

Evaluate the integrals $$ \int_{1}^{4}\left(3 x^{2}-\frac{x^{3}}{4}\right) d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{753}{16}\).

Step 1: Split the Integral

We begin by splitting the integral into two separate components: one for each term within the integrand. We have:\[ \int_{1}^{4}\left(3x^{2} - \frac{x^{3}}{4}\right) \, dx = \int_{1}^{4}3x^{2} \, dx - \int_{1}^{4}\frac{x^{3}}{4} \, dx \]

Step 2: Evaluate the First Integral

We evaluate the first integral, \(\int_{1}^{4}3x^{2} \, dx\). Using the power rule for integration, we find:\[ \int 3x^{2} \, dx = 3 \cdot \frac{x^{3}}{3} = x^{3} \] Thus,\[ \int_{1}^{4}3x^{2} \, dx = [x^{3}]_{1}^{4} = 4^{3} - 1^{3} = 64 - 1 = 63 \]

Step 3: Evaluate the Second Integral

Next, we evaluate the integral \(\int_{1}^{4}\frac{x^{3}}{4} \, dx\). We can factor out the constant \(\frac{1}{4}\) to simplify:\[ \frac{1}{4} \int x^{3} \, dx = \frac{1}{4} \cdot \frac{x^{4}}{4} = \frac{x^{4}}{16} \] Thus,\[ \int_{1}^{4}\frac{x^{3}}{4} \, dx = \left[\frac{x^{4}}{16}\right]_{1}^{4} = \frac{4^{4}}{16} - \frac{1^{4}}{16} = \frac{256}{16} - \frac{1}{16} = 16 - \frac{1}{16} = \frac{255}{16} \]

Step 4: Combine the Results

Finally, we combine the results from Step 2 and Step 3 to find the value of the original integral:\[ 63 - \frac{255}{16} = \frac{1008}{16} - \frac{255}{16} = \frac{753}{16} \]

Key Concepts

Integration by Parts

Integration by parts is a useful technique for solving integrals that are products of functions. This method is based on the product rule for differentiation, and it helps us to split a complicated integral into simpler parts that are easier to solve.
To apply integration by parts, we typically use the formula:

• \[ \int u \, dv = uv - \int v \, du \]

Here, \(u\) and \(dv\) are parts of the original integral we choose. This technique is particularly helpful when we have an integral involving a product of an algebraic function and another function like a logarithmic or exponential function.
In our exercise, integration by parts isn't directly used because the function is a polynomial that is more straightforwardly decomposed by simpler methods. However, understanding this technique is crucial for more advanced integrals. It's a powerful tool to simplify integrals that might otherwise seem challenging.

Power Rule for Integration

The power rule for integration is a fundamental tool in calculus. It simplifies the process of finding integrals for functions that are powers of \(x\).
The formula for the power rule is simple to remember:

• If \( n eq -1 \), then \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

The constant \(C\) represents the constant of integration, typically not needed for definite integrals. For definite integrals, like in our exercise, we evaluate the antiderivative at the range provided and subtract.
In the given solution, the power rule is applied to both terms inside the integral. For \(3x^2\), the rule quickly helps us determine that the antiderivative is \(x^3\). This process is essential for breaking down polynomial integrals into solvable components.
For students, remember to increase the exponent by one and divide by the new exponent. It's a simple yet effective way to handle polynomial integrals, making it an indispensable skill in calculus.

Splitting Integrals

Splitting integrals is a straightforward but incredibly helpful concept in integration. When faced with an integral containing a sum or difference of functions, it's often best to separate them into individual integrals.
This was the first step in our original problem, where the integral of \(3x^2 - \frac{x^3}{4}\) was split into two parts:

• \[ \int 3x^2 \, dx \]
• \[ - \int \frac{x^3}{4} \, dx \]

By splitting up complex expressions, we can deal with each term separately, often utilizing the power rule to solve for their antiderivatives. This method reduces the potential for errors that can occur when working with more complicated expressions as a whole.
After solving each integral, we can recombine the results to get the final answer for the original problem. It's a technique that prioritizes clarity and simplicity, allowing us to approach each component of an integral individually and ensuring accuracy in our computations.