Set up and evaluate a definite integral for the area of the region circumscribed by the graphs of y=3xy=3x and y=7−xy=7-x.


To find the area of the region circumscribed by the graphs of \( y = 3x \) and \( y = 7 - x \), we need to follow these steps:

### Step 1: Find the points of intersection
To determine the limits of integration, we must first find where the two graphs intersect by setting \( y = 3x \) equal to \( y = 7 - x \):

\[
3x = 7 - x
\]
\[
3x + x = 7
\]
\[
4x = 7
\]
\[
x = \frac{7}{4}
\]

Thus, the two graphs intersect at \( x = \frac{7}{4} \). Since we are looking for the area bounded by these curves, we also check the intersection when \( x = 0 \), because both lines pass through the point \( (0, 0) \).

### Step 2: Set up the definite integral
The area between the two curves can be found by integrating the difference of the upper function \( y = 7 - x \) and the lower function \( y = 3x \) over the interval \( [0, \frac{7}{4}] \).

Thus, the area \( A \) is given by:

\[
A = \int_0^{\frac{7}{4}} \left( (7 - x) - (3x) \right) dx
\]
Simplifying the integrand:

\[
A = \int_0^{\frac{7}{4}} \left( 7 - x - 3x \right) dx = \int_0^{\frac{7}{4}} \left( 7 - 4x \right) dx
\]

### Step 3: Evaluate the integral
Now we compute the definite integral:

\[
A = \int_0^{\frac{7}{4}} (7 - 4x) dx
\]

First, integrate each term separately:

\[
\int (7 - 4x) dx = 7x - 2x^2
\]

Now, apply the limits of integration from \( 0 \) to \( \frac{7}{4} \):

\[
A = \left[ 7x - 2x^2 \right]_0^{\frac{7}{4}}
\]

Substitute the upper limit \( x = \frac{7}{4} \):

\[
7\left(\frac{7}{4}\right) - 2\left(\frac{7}{4}\right)^2 = \frac{49}{4} - 2\left(\frac{49}{16}\right) = \frac{49}{4} - \frac{98}{16} = \frac{49}{4} - \frac{49}{8} = \frac{98}{8} - \frac{49}{8} = \frac{49}{8}
\]

Substitute the lower limit \( x = 0 \) (which yields 0):

\[
7(0) - 2(0)^2 = 0
\]

Thus, the total area is:

\[
A = \frac{49}{8}
\]

### Final Answer:
The area of the region bounded by the curves is \( \frac{49}{8} \) square units.

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Would you like more details on any step?

### Related Questions:
1. How can we find the volume of a solid generated by rotating the region around the x-axis?
2. How do we compute the area if the curves are reversed, with \( y = 3x \) as the upper function?
3. Can you explain how to solve the same problem using numerical integration?
4. What is the meaning of the integrand in terms of geometry?
5. How would the problem change if we integrated over a different interval?

### Tip:
Always graph the functions to verify the limits and ensure you are subtracting the correct curve from the other.

Set up and evaluate a definite integral for the area of the region circumscribed by the graphs of y = 3x and y = 7 - x.

Explore how to calculate the area of the region bounded by the linear functions y = 3x and y = 7 - x. This problem involves setting up and evaluating a definite integral using the Fundamental Theorem of Calculus. Suitable for high school students in Grades 11-12.

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