R
is the region bounded by the functions
f
(
x
)
=
2/
x
and
g
(
x
)
=
−
x/
5
+
7/
5
.
Find the area
A
of
R
.
Enter answer using exact values.

To find the area \(A\) of the region \(R\) bounded by the functions \(f(x) = \frac{2}{x}\) and \(g(x) = -\frac{x}{5} + \frac{7}{5}\), we need to follow these steps:

1. **Find the points of intersection** of \(f(x)\) and \(g(x)\).
2. **Set up the integral** that represents the area between these curves over the interval determined by the intersection points.
3. **Evaluate the integral** to find the exact area.

### Step 1: Find the Points of Intersection

To find the points of intersection, we set \(f(x) = g(x)\):

\[
\frac{2}{x} = -\frac{x}{5} + \frac{7}{5}
\]

Multiply both sides by \(5x\) (to eliminate the fraction):

\[
5x \cdot \frac{2}{x} = 5x \left(-\frac{x}{5} + \frac{7}{5}\right)
\]

Simplify:

\[
10 = -x^2 + 7x
\]

Rearrange the equation to form a standard quadratic equation:

\[
x^2 - 7x + 10 = 0
\]

Factor the quadratic equation:

\[
(x - 2)(x - 5) = 0
\]

So, the solutions are:

\[
x = 2 \quad \text{and} \quad x = 5
\]

These are the \(x\)-coordinates of the points where the curves intersect.

### Step 2: Set Up the Integral

Now we set up the integral to find the area between the curves from \(x = 2\) to \(x = 5\).

Since \(f(x) = \frac{2}{x}\) is above \(g(x) = -\frac{x}{5} + \frac{7}{5}\) in this interval, the area \(A\) is given by:

\[
A = \int_{2}^{5} \left( \frac{2}{x} - \left(-\frac{x}{5} + \frac{7}{5}\right) \right) dx
\]

Simplify the integrand:

\[
A = \int_{2}^{5} \left( \frac{2}{x} + \frac{x}{5} - \frac{7}{5} \right) dx
\]

### Step 3: Evaluate the Integral

Evaluate each term separately:

\[
\int \frac{2}{x} \, dx = 2 \ln|x|, \quad \int \frac{x}{5} \, dx = \frac{x^2}{10}, \quad \int \frac{7}{5} \, dx = \frac{7x}{5}
\]

Now, compute the definite integral:

\[
A = \left[ 2 \ln|x| + \frac{x^2}{10} - \frac{7x}{5} \right]_{2}^{5}
\]

Evaluate at the upper limit \(x = 5\) and the lower limit \(x = 2\):

\[
A = \left(2 \ln 5 + \frac{5^2}{10} - \frac{7 \cdot 5}{5}\right) - \left(2 \ln 2 + \frac{2^2}{10} - \frac{7 \cdot 2}{5}\right)
\]

Simplify each part:

\[
A = \left(2 \ln 5 + \frac{25}{10} - 7\right) - \left(2 \ln 2 + \frac{4}{10} - \frac{14}{5}\right)
\]

\[
A = \left(2 \ln 5 + 2.5 - 7\right) - \left(2 \ln 2 + 0.4 - 2.8\right)
\]

\[
A = \left(2 \ln 5 - 4.5\right) - \left(2 \ln 2 - 2.4\right)
\]

\[
A = 2 \ln 5 - 4.5 - 2 \ln 2 + 2.4
\]

Combine like terms:

\[
A = 2(\ln 5 - \ln 2) - 2.1
\]

Using the logarithm property \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\):

\[
A = 2 \ln \left(\frac{5}{2}\right) - 2.1
\]

Thus, the exact area \(A\) is:

\[
A = 2 \ln \left(\frac{5}{2}\right) - 2.1
\]

### Conclusion

The area \(A\) of the region \(R\) is \(2 \ln \left(\frac{5}{2}\right) - 2.1\).

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you find the area between two curves that intersect at more than two points?
2. What is the significance of using logarithmic properties in simplifying integrals?
3. How do you handle integrals with absolute values in their limits?
4. What are some common mistakes to avoid when setting up integrals for areas?
5. How do the shapes of the curves affect the setup of the integral for the area?

**Tip:** Always check the orientation of the curves before setting up the integral to ensure you're calculating the area correctly.

Learn how to find the area of the region bounded by the functions f(x) = 2/x and g(x) = -x/5 + 7/5 using integration. This problem involves solving for the points of intersection, setting up and evaluating the integral, and applying the Fundamental Theorem of Calculus. Suitable for advanced high school students.

Math Problem Statement

R is the region bounded by the functions f ( x )

2/ x and g ( x )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation