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The integral to compute is:

\[
\int_2^3 -7r^{1/2} \, dr
\]

### Step 1: Apply the Power Rule for Integration
The general form of the integral of a power function is:

\[
\int r^n \, dr = \frac{r^{n+1}}{n+1} + C
\]

Here, \( r^{1/2} \) can be integrated using this rule. So, the first step is to integrate \( r^{1/2} \).

\[
\int r^{1/2} \, dr = \frac{r^{(1/2) + 1}}{(1/2) + 1} = \frac{r^{3/2}}{3/2} = \frac{2}{3} r^{3/2}
\]

Now, we multiply this result by \(-7\), since the original integral has \(-7\) as a constant factor:

\[
\int -7r^{1/2} \, dr = -7 \cdot \frac{2}{3} r^{3/2} = -\frac{14}{3} r^{3/2}
\]

### Step 2: Evaluate the Definite Integral
Now, evaluate this expression at the limits \( r = 3 \) and \( r = 2 \):

\[
-\frac{14}{3} \left[ r^{3/2} \right]_2^3 = -\frac{14}{3} \left( 3^{3/2} - 2^{3/2} \right)
\]

### Step 3: Calculate the Values
1. \( 3^{3/2} = \sqrt{3^3} = \sqrt{27} \approx 5.196 \)
2. \( 2^{3/2} = \sqrt{2^3} = \sqrt{8} \approx 2.828 \)

Now, substitute these into the expression:

\[
-\frac{14}{3} \left( 5.196 - 2.828 \right) = -\frac{14}{3} \times 2.368 = -\frac{14 \times 2.368}{3}
\]

### Step 4: Final Calculation

\[
-\frac{14 \times 2.368}{3} \approx -\frac{33.152}{3} \approx -11.05
\]

Thus, the value of the integral is approximately:

\[
\boxed{-11.05}
\]

Would you like more details on any step or clarification?

Here are five related questions for further understanding:
1. How do you apply the power rule for different exponents?
2. How would the solution change if the bounds of integration were different?
3. What happens if we change the exponent of \(r\) in the original integrand?
4. How would you approach integrating an expression with a fractional exponent like \(r^{3/2}\)?
5. Why is evaluating definite integrals important in real-world applications?

**Tip:** When working with definite integrals, make sure to carefully evaluate the function at both the upper and lower limits to avoid common calculation errors.

Compute the value of the integral from 2 to 3 of -7r^(1/2) dr.

This page explains how to compute the definite integral of -7r^(1/2) from 2 to 3 using the power rule for integration. Includes step-by-step breakdown of the solution, involving calculus techniques like integration and evaluating the definite integral.

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