Question

In Exercises \(47-56,\) use graphs, your knowledge of area, and the fact that \(\quad\) $$\int_{0}^{1} x^{3} d x=\frac{1}{4}$$ to evaluate the integral. $$\int_{0}^{2}\left(\frac{x}{2}\right)^{3} d x$$

Short answer

The value of the integral \( \int_{0}^{2}\left(\frac{x}{2}\right)^{3} d x \) is \(\frac{1}{16}\).

Step-by-step solution

Identify the Integral to be Solved

First, it's important to identify the integral that needs to be solved: \[\int_{0}^{2}\left(\frac{x}{2}\right)^{3} d x\] This is the definite integral of the function \( \left(\frac{x}{2}\right)^{3} \) with respect to \( x \) in the interval [0, 2]

Apply Constant Multiple Rule

The constant multiple rule states that the integral of a constant times a function is equal to the constant times the integral of the function. Apply this rule to simplify the given integral: \[\int_{0}^{2}\left(\frac{x}{2}\right)^{3} d x = \frac{1}{8} \int_{0}^{2} x^{3} d x\]

Change of Limits scaling

Note that our integral now is similar to the given integral \( \int_{0}^{1} x^{3} d x = \frac{1}{4} \). However, our limit is 0 to 2, instead of 0 to 1. Fortunately, we can account for this by scaling the change of integral limits: \[\int_{0}^{2} x^{3} d x = 2 \int_{0}^{1} x^{3} d x\]

Compute Final Answer

We now substitute the given integral value into our expression to get the final answer: \[\frac{1}{8} \int_{0}^{2} x^{3} d x = \frac{1}{8} * 2 * \frac{1}{4} = \frac{1}{16}\]