Question:

Let $f(x) = x^2 - 3x$ and $g(x) = 2x - 1$ be two functions defined for $0 \leq x \leq 4$. Find the area of the region bounded by the curves $f(x)$ and $g(x)$.

Answer:

To find the area of the region bounded by the curves $f(x)$ and $g(x)$, we need to determine the points of intersection between the two curves first.

Setting $f(x) = g(x)$, we have:

$x^2 - 3x = 2x - 1$

Simplifying the equation, we get:

$x^2 - 5x + 1 = 0$

We can solve this quadratic equation by factoring or using the quadratic formula. However, upon inspection, we can see that there are no real solutions for this equation. Thus, the curves $f(x)$ and $g(x)$ do not intersect in the given interval $0 \leq x \leq 4$.

Therefore, the area bounded by the curves is equal to the area under $f(x)$ minus the area under $g(x)$.

Let's first find the area under $f(x)$:

Using the power rule of integration, we can integrate term by term:

Next, let's find the area under $g(x)$:

Again, we can integrate term by term:

Now, we can find the area bounded by the curves:

Therefore, the area of the region bounded by the curves $f(x)$ and $g(x)$ is $-\frac{44}{3}$ square units.