Animated Solution for Mathematics - Definite Integration: Let $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral ${\int }_0^1[-8x^2+6x-1]dx$ is equal to

Imagine you are standing before a complex, shifting landscape. You have a quadratic function, $f(x)=-8x^2+6x-1$, and you are asked to find the area under its 'staircase' version, the greatest integer function $[f(x)]$.

When we see $[f(x)]$, we shouldn't panic. Instead, we should see a staircase where $f(x)$ is the smooth, continuous path and the greatest integer function is the rigid, stepped structure that follows it.

Our goal is to calculate the area under this staircase from $x=0$ to $x=1$.

First, let us understand the behavior of our quadratic function $f(x)=-8x^2+6x-1$. This is an inverted parabola.

To understand its journey, we find its vertex using:

At this peak, the value is $f(\frac{3}{8})=\frac{1}{8}$. At the boundaries, $f(0)=-1$ and $f(1)=-3$. This tells us that as $x$ moves from $0$ to $1$, the function climbs from $-1$ to $1/8$ and then descends to $-3$.

The greatest integer function $[f(x)]$ only changes its value when $f(x)$ crosses an integer. We need to find exactly where this happens.

We set $f(x)=0$, which gives us $(4x-1)(2x-1)=0$, yielding roots at $x=1/4$ and $x=1/2$. In this interval, $f(x)$ is between $0$ and $1/8$, so $[f(x)]=0$.

Next, we look for $f(x)=-1$. Solving $-8x^2+6x-1=-1$ gives $x=0$ and $x=3/4$.

Finally, for $f(x)=-2$, we solve $-8x^2+6x-1=-2$, or $8x^2-6x-1=0$. Using the quadratic formula, we find: We take the positive root for the right side of the parabola.

Now, the integral $I={\int }_0^1[f(x)]dx$ transforms into a sum of simple rectangular areas:

Each integral is just the height of the step multiplied by the width of the interval. The first three parts are straightforward: $I_1=-1/4$, $I_2=0$, and $I_3=-1/4$. Summing these gives $-1/2$.

For the fourth part:

For the final part:

Adding these together:

Converting $-1/2$ to $-4/8$, we get:

This is the beauty of calculus. We took a complex, discontinuous function and, by understanding its geometric soul, broke it down into simple, manageable pieces. You have successfully navigated the staircase!