*y* = *x*,*y* =(1/2)*x*,   *x* = 25

Find the area of the region.

To find the area of the region enclosed by the lines \( y = x \), \( y = \frac{1}{2}x \), and the vertical line \( x = 25 \), we'll follow these steps:

### Step 1: Identify the Intersection Points

The curves \( y = x \) and \( y = \frac{1}{2}x \) intersect at the origin \((0, 0)\), since both pass through this point. The vertical line \( x = 25 \) intersects both lines at:

- For \( y = x \): \( (25, 25) \)
- For \( y = \frac{1}{2}x \): \( (25, \frac{1}{2} \times 25) = (25, 12.5) \)

### Step 2: Set Up the Integral for the Area

The area of the region is the area between the two curves from \( x = 0 \) to \( x = 25 \). The region is bounded by the curves:

- Top curve: \( y = x \)
- Bottom curve: \( y = \frac{1}{2}x \)

The area \( A \) is given by the integral of the difference between the two curves:

\[
A = \int_{0}^{25} \left( x - \frac{1}{2}x \right) \, dx
\]

### Step 3: Evaluate the Integral

Simplify the integrand:

\[
A = \int_{0}^{25} \frac{1}{2}x \, dx
\]

Now, integrate:

\[
A = \frac{1}{2} \int_{0}^{25} x \, dx = \frac{1}{2} \left[ \frac{x^2}{2} \right]_{0}^{25}
\]

Evaluate at the bounds:

\[
A = \frac{1}{2} \left[ \frac{25^2}{2} - \frac{0^2}{2} \right] = \frac{1}{2} \times \frac{625}{2} = \frac{625}{4}
\]

Thus, the area of the region is \( \boxed{156.25} \).

Do you want details or have any questions?

### Related Questions:
1. What would the area be if the vertical line were \( x = 50 \)?
2. How does the area change if \( y = \frac{1}{3}x \) replaces \( y = \frac{1}{2}x \)?
3. What if the line \( y = -\frac{1}{2}x \) were included in the region?
4. How does the area calculation change if the region is bounded by \( x = 10 \) instead of \( x = 25 \)?
5. What is the geometric significance of the difference in slope between the two lines?

**Tip:** For simple linear regions, understanding the geometry of the problem can often lead to quicker solutions, sometimes even avoiding integration entirely by using basic geometric formulas.

Learn how to calculate the area between curves y = x and y = (1/2)x bounded by x = 25 using integral calculus. Detailed step-by-step solution provided.

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