Find the area of the region described in the following exercises.


The region bounded by y=4x+4, y=6x+6, and x=4

Verified step by step guidance

1. Step 1: Identify the boundaries of the region. The region is bounded by the lines y = 4x + 4, y = 6x + 6, and x = 4. These equations describe straight lines, and the intersection points will help define the vertices of the region.
2. Step 2: Find the intersection points of the lines. To find where y = 4x + 4 intersects y = 6x + 6, set the two equations equal to each other: ${\mathrm{4x}}^{+}4={\mathrm{6x}}^{+}6$. Solve for x, then substitute the x-value into either equation to find the corresponding y-coordinate.
3. Step 3: Determine the points where the lines intersect the vertical line x = 4. Substitute x = 4 into y = 4x + 4 and y = 6x + 6 to find the y-coordinates of the intersection points. This will give you two points on the vertical line.
4. Step 4: Sketch the region. Plot the lines y = 4x + 4, y = 6x + 6, and x = 4 on a graph. Mark the intersection points found in Steps 2 and 3. The region is a triangular area enclosed by these boundaries.
5. Step 5: Calculate the area of the triangular region. Use the formula for the area of a triangle: $Area=\frac{1}{2}\times (\mathrm{base}\times \mathrm{height})$. Determine the base and height of the triangle from the graph or the coordinates of the vertices, and substitute these values into the formula.

Key Concepts

Linear Equations

Linear equations represent straight lines in a coordinate plane. The equations given, y=4x+4 and y=6x+6, describe two lines with different slopes. Understanding how to graph these lines is essential for visualizing the bounded region and determining the area between them.

Area Between Curves

The area between two curves can be found by integrating the difference of the functions over a specified interval. In this case, the area between the lines y=4x+4 and y=6x+6 from the leftmost intersection point to x=4 needs to be calculated. This involves setting up the integral of the upper function minus the lower function.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points on the x-axis. They provide a numerical value representing the total area, which is crucial for solving the problem. The limits of integration will be determined by the points where the two lines intersect and the vertical line x=4.