To find the area between the parabola $y = x^2$ and the line $y = 2x + 3$, we need to follow these steps:

Step 1: Find the points of intersection

We first set the equations equal to each other to find the $x$-coordinates where the parabola and the line intersect:

$x^2 = 2x + 3$

Rearranging this equation:

$x^2 - 2x - 3 = 0$

We solve this quadratic equation by factoring:

$(x - 3)(x + 1) = 0$

So, the points of intersection are:

$x = 3 \quad \text{and} \quad x = -1$

Thus, the points of intersection are $x = -1$ and $x = 3$.

Step 2: Set up the integral for the area

The area between the curves is given by the integral of the difference between the two functions from $x = -1$ to $x = 3$. The area is:

$A = \int_{-1}^{3} \left[ (2x + 3) - x^2 \right] \, dx$

Step 3: Simplify and integrate

We now simplify the integrand:

$A = \int_{-1}^{3} (2x + 3 - x^2) \, dx$

Now, we compute the integral of each term:

$\int 2x \, dx = x^2$ $\int 3 \, dx = 3x$ $\int -x^2 \, dx = -\frac{x^3}{3}$

Thus, the integral becomes:

$A = \left[ x^2 + 3x - \frac{x^3}{3} \right]_{-1}^{3}$

Step 4: Evaluate the integral

Now, evaluate the expression at the limits $x = 3$ and $x = -1$:

At $x = 3$:

$\left( 3^2 + 3(3) - \frac{3^3}{3} \right) = 9 + 9 - 9 = 9$

At $x = -1$:

$\left( (-1)^2 + 3(-1) - \frac{(-1)^3}{3} \right) = 1 - 3 + \frac{1}{3} = -2 + \frac{1}{3} = -\frac{5}{3}$

Step 5: Subtract the results

The area is:

$A = 9 - \left( -\frac{5}{3} \right) = 9 + \frac{5}{3} = \frac{27}{3} + \frac{5}{3} = \frac{32}{3}$

Final Answer:

The area between the parabola and the line is $\frac{32}{3}$ square units.

---

Would you like any more details on the solution, or do you have additional questions about this problem?

Here are some related questions to explore further:

1. How can we find the area between curves using integration for other types of functions?
2. How do you determine whether the integral is set up correctly for finding the area?
3. How does the concept of definite integrals relate to finding areas under curves?
4. How would the solution change if the curves were not polynomial functions?
5. What would the area be if we considered the region between the curves for a different range of $x$?

Tip: Always check that the integrand represents the "top" curve minus the "bottom" curve when finding the area between two functions.