Calculating the Area of a Figure Bounded by Specific Lines in the Coordinate Plane

When working in the coordinate plane, it's essential to understand how to calculate the area of a figure bounded by specific lines. This involves both geometric methods and integration in calculus. The example used here involves finding the area of a trapezoid defined by the lines x 1, y 3, y 0, and y x - 4.

Geometric Approach: Calculating the Area of a Trapezoid

The first step is to visualize and understand the figure. The figure bounded by the lines x 1, y 3, y 0, and y x - 4 is a trapezoid. To find its area, we need to identify the bases and the height.

Identifying the Bases and Height

To determine the bases, we need to find the points where the lines intersect. The line y 3 intersects the line y x - 4 at the point where 3 x - 4, which gives us x 7. Similarly, the line y 0 intersects the line y x - 4 at the point where 0 x - 4, which gives us x 4. This means the bases of the trapezoid are the segments of the line y 0 from x 4 to x 1, and the segments of the line y 3 from x 7 to x 1.

Therefore, the length of one base is 7 - 4 3, and the other base is 7 - 1 6. The height of the trapezoid is the vertical distance between the lines y 3 and y 0, which is 3.

Calculating the Area

The formula for the area of a trapezoid is given by:

Area (frac{(base1 base2) times height}{2})

Substituting the values we have:

(Area frac{(3 6) times 3}{2} frac{27}{2} 13.5)

Calculus Approach: Using Integration to Find the Area

Another method to solve this problem is by using integration. The area under the region can be found by integrating the function y x - 4 from the intersection points x 1 to x 4 (where it crosses y 0), and then adding the area from x 4 to x 7 (where it crosses y 3). However, a more precise approach is to use the line (y 3 - (x - 4)) for the upper bound within the region.

Setting Up the Integral

The area can be calculated by evaluating the definite integral:

[ int_{0}^{3} (3 - x 4) , dy ]

This simplifies to:

[ int_{0}^{3} (y - (x - 4)) , dy ]

Which becomes:

[ int_{0}^{3} (y - x 4) , dy ]

Since the area is defined by vertical slices, we can use:

[ int_{1}^{4} (3 - x 4) , dx ]

For the definite integral:

[ int_{1}^{4} (3 - y - 4) , dy ]

[ int_{1}^{4} (y 3) , dy - int_{1}^{4} (y) , dy ]

[ left[ frac{y^2}{2} 3y right]_{1}^{4} - left[ frac{y^2}{2} right]_{1}^{4} ]

[ left( frac{16}{2} 3 cdot 4 right) - left( frac{1}{2} 3 right) - left( frac{16}{2} - frac{1}{2} right) ]

[ (8 12) - (1.5 3) - (8 - 0.5) ]

[ 20 - 4.5 - 7.5 8 ]

Thus, the area is:

[ frac{27}{2} 13.5 ]

Conclusion

Both the geometric and calculus approaches yield the same result of 13.5 square units. By understanding the properties of the trapezoid and applying integration, you can accurately calculate the area of figures defined in the coordinate plane.

Keywords:

coordinate plane, trapezoid, integral calculus