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To find the area between the curves \( y = 9x \) and \( y = x^2 - 10 \) between \( x = -2 \) and \( x = 3 \), we can follow these steps:

### Step 1: Set up the integral for the area.
The area between two curves is given by the formula:
\[
A = \int_{a}^{b} \left( f(x) - g(x) \right) \, dx
\]
where \( f(x) \) is the upper curve and \( g(x) \) is the lower curve.

In this case, the line \( y = 9x \) will be the upper curve, and the parabola \( y = x^2 - 10 \) will be the lower curve. So, the area between the curves is:
\[
A = \int_{-2}^{3} \left( 9x - (x^2 - 10) \right) dx
\]

### Step 2: Simplify the expression inside the integral.
\[
9x - (x^2 - 10) = 9x - x^2 + 10
\]

Thus, the integral becomes:
\[
A = \int_{-2}^{3} \left( -x^2 + 9x + 10 \right) dx
\]

### Step 3: Compute the integral.
Now we integrate term by term:
\[
\int \left( -x^2 + 9x + 10 \right) dx = -\frac{x^3}{3} + \frac{9x^2}{2} + 10x
\]

### Step 4: Evaluate the definite integral.
Now we evaluate from \( x = -2 \) to \( x = 3 \):
\[
A = \left[ -\frac{x^3}{3} + \frac{9x^2}{2} + 10x \right]_{-2}^{3}
\]

First, calculate at \( x = 3 \):
\[
-\frac{3^3}{3} + \frac{9(3^2)}{2} + 10(3) = -\frac{27}{3} + \frac{9 \times 9}{2} + 30 = -9 + 40.5 + 30 = 61.5
\]

Next, calculate at \( x = -2 \):
\[
-\frac{(-2)^3}{3} + \frac{9(-2)^2}{2} + 10(-2) = -\frac{-8}{3} + \frac{9 \times 4}{2} - 20 = \frac{8}{3} + 18 - 20 = \frac{8}{3} - 2 = \frac{2}{3}
\]

### Step 5: Subtract to find the area.
Now subtract the two results:
\[
A = 61.5 - \frac{2}{3} = \frac{184.5}{3} - \frac{2}{3} = \frac{182.5}{3} = 60.83 \, \text{square units}
\]

Thus, the area between the curves is approximately \( 60.83 \) square units.

Would you like further clarification or details? 

Here are five related questions you might find helpful:
1. How do we determine which curve is the upper or lower one?
2. What are the general steps to finding the area between any two curves?
3. How would the integral change if the limits of integration were reversed?
4. Can we use geometry to check the answer for linear and quadratic curves?
5. What happens if the curves intersect within the bounds?

**Tip:** Always make sure to simplify the expressions inside the integral before computing. This helps avoid errors in evaluating the definite integral.

Find the area between the curves x = -2, x = 3, y = 9x, y = x^2 - 10.

Learn how to calculate the area between the curves y = 9x and y = x^2 - 10 between x = -2 and x = 3 using definite integrals. This solution covers key concepts in calculus, including the area between two curves and integral calculus.

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