Question

\(\int\limits_0^2 {(x - {x^3})dx} \)represents the area under the curve \(y = x - {x^3}\) from \(0\;to\;2\).

Short answer

The given statement is TRUE.

Step-by-step solution

Step 1: Explanation with the help of graph

As seen in the image, the graph of the curve is divided by the x-axis at \(x = 1\).

The area of A is defined by \(\int\limits_0^2 {(x - {x^3})dx} \).

Meanwhile, the area of B is defined by \(\int\limits_1^2 {(x - {x^3})dx} \). Note that the integral of B is expected to have negative answer, but this is expected because the area is wholly below the x-axis

Step 2: Calculating the area of the given function

Calculating the area of given function, we now have,

\(\int\limits_0^1 {(x - {x^3})dx} + \left| {\int\limits_1^2 {(x - {x^3})dx} } \right|\)

The function is continuous on (0, 2) so it is safe to say that this is true.