Problem 8 \(\int_{2}^{4} x^{2} d x\)... [FREE SOLUTION]

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Chapter 7: Problem 8

\(\int_{2}^{4} x^{2} d x\)

Short Answer

Expert verified

The value of the definite integral is \( \frac{56}{3} \).

Step by step solution

Define the Integral

Identify the integral to be evaluated: \[ \int_{2}^{4} x^{2} \, dx \]

Find the Antiderivative

Find the antiderivative of the integrand. The antiderivative of \(x^{2}\) is \(\frac{x^{3}}{3}\).

Evaluate the Antiderivative at the Bounds

Evaluate the antiderivative at the upper and lower bounds of the integral:\[ \left. \frac{x^{3}}{3} \right|_{2}^{4} \]

Substitute the Bounds

Substitute the upper bound (4) and the lower bound (2) into the antiderivative:\[ \frac{4^{3}}{3} - \frac{2^{3}}{3} \]

Simplify the Expression

Simplify the expression to find the value of the definite integral:\[ \frac{64}{3} - \frac{8}{3} = \frac{56}{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative

To solve a definite integral, we first need to find the antiderivative of the function we are integrating. The antiderivative is essentially the reverse process of taking a derivative. In this specific exercise, we have the function \(x^{2}\).
The antiderivative of \(x^{2}\) is found using the rule for power functions: \[ \frac{x^{n+1}}{n+1} \]
Applying this rule to \(x^{2}\), we get \[ \frac{x^{3}}{3} \]
This function, \( \frac{x^{3}}{3} \), is the antiderivative of \( x^{2} \). Finding the antiderivative is the first and crucial step before proceeding to evaluate the definite integral.

Evaluation of Bounds

Once the antiderivative is determined, the next step is to evaluate it at the given bounds of the integral. Here, our bounds are from 2 to 4.
We use the notation: \[ \[\begin{equation} \bigg[ \frac{x^{3}}{3} \bigg]_{2}^{4} \end{equation}\] \]
This means we need to substitute the upper bound (4) and the lower bound (2) into our antiderivative function:

Substitute the upper bound: \ \frac{4^{3}}{3} \

Substitute the lower bound: \ \frac{2^{3}}{3} \

Then we subtract the result of the lower bound from the result of the upper bound:
\[ \frac{4^{3}}{3} - \frac{2^{3}}{3} \]

Simplification

After substituting the bounds into the antiderivative function, we get:
\ \frac{64}{3} - \frac{8}{3} \
At this point, we need to simplify the expression. Simplification involves performing basic arithmetic operations.
Since both terms are fractions with the same denominator, we can directly subtract the numerators:

\ 64 - 8 = 56 \

So, the expression simplifies to:
\[ \frac{56}{3} \]
This final value, \ \frac{56}{3} \, is the result of the definite integral \ \[\begin{equation} \bigg(\textstyle \big[ \frac{x^2}{2} \big]_{2}^{4} \bigg) \end{equation}\] \.

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91影视

\(\int_{2}^{4} x^{2} d x\)

Short Answer

Step by step solution

Define the Integral

Find the Antiderivative

Evaluate the Antiderivative at the Bounds

Substitute the Bounds

Simplify the Expression

Key Concepts

Antiderivative

Evaluation of Bounds

Simplification

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