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. $$\int_{0}^{1}\left(1-x^{3}\right) d x$$

Short answer

The evaluated value of the given integral, \(\int_{0}^{1}(1 - x^{3}) dx\), is \(\frac{3}{4}\).

Step-by-step solution

Recognizing the problem

The given integral is \(\int_{0}^{1}(1-x^{3}) dx\). This is similar to the integral \(\int_{0}^{1} x^{3} dx\), which is given as \(\frac{1}{4}\). The difference is, in the given integral, instead of \(x^{3}\), we have \((1- x^{3})\). This subtraction can be dealt with by considering the difference as a separate integral.

Separating the integral

The properties of definite integrals allow for the breaking apart of composite functions into separate integrals. This is particularly useful in this problem as we have already been given the value of one of those integrals. Hence, the integral can be written as \(\int_{0}^{1}(1-x^{3}) dx = \int_{0}^{1} dx - \int_{0}^{1} x^{3} dx\)

Solving the integral

Now, we can evaluate both integrals separately. The integral \(\int_{0}^{1} dx\) is the area under the curve of the function \(y=1\) from 0 to 1. So this integral is simply 1. The integral \(\int_{0}^{1} x^{3} dx\) is given as \(\frac{1}{4}\). So, using those values, the solution to the given integral is: \(1 - \frac{1}{4} = \frac{3}{4}\).