Calculate the total area of the region described. Do not count area beneath the x-axis as negative.Bounded by the line y = 6x, the x-axis, and the lines x = 4 and x = 5
a = ∫[4,5] 6x dx = 27
or, since the area is just a trapezoid of height 1,
a = (24+30)/2 = 27
To calculate the total area of the region described, we need to break it down into smaller sections.
Step 1: Find the area between the line y = 6x and the x-axis.
This region is a triangle. To find the area of a triangle, we use the formula: Area = (base * height)/2.
In this case, the base is the x-axis, and the height is the y-coordinate of the line y = 6x.
Since the line intersects the x-axis at y = 0, the height of the triangle is 0. The base is the difference in x-values between the given x-axis boundaries: 5 - 4 = 1.
Therefore, the area of this triangle is 0.
Step 2: Find the area between the line y = 6x and the lines x = 4 and x = 5.
This region is a trapezoid. To find the area of a trapezoid, we use the formula: Area = (sum of parallel sides * height)/2.
In this case, the parallel sides are the y-coordinates of the line y = 6x and the x-axis, and the height is the difference in x-values between the given x-axis boundaries: 5 - 4 = 1.
The y-coordinate of the line y = 6x when x = 4 is 6 * 4 = 24. The y-coordinate of the line y = 6x when x = 5 is 6 * 5 = 30.
Therefore, the sum of parallel sides is 24 + 0 = 24, and the area of this trapezoid is (24 * 1)/2 = 12.
Step 3: Calculate the total area.
To find the total area, we add up the areas of the triangle and the trapezoid.
Total area = 0 (triangle) + 12 (trapezoid) = 12 square units.
So, the total area of the region described is 12 square units.
To find the total area of the region described, we need to calculate the area of each individual shape within the bounded region and then add them together.
First, let's identify the shapes within the bounded region.
1. Triangle: The region below the line y = 6x and above the x-axis for x values between x = 4 and x = 5.
2. Rectangle: The region between the x-axis and the line y = 6x for x values between x = 4 and x = 5.
Now, let's calculate the area of each shape.
1. Triangle: The base of the triangle is the difference in x-coordinates between x = 4 and x = 5, which is 5 - 4 = 1. The height of the triangle is the y-coordinate at the top vertex, which is 6(4) = 24. Therefore, the area of the triangle is (1/2) * base * height = (1/2) * 1 * 24 = 12.
2. Rectangle: The length of the rectangle is again the difference in x-coordinates between x = 4 and x = 5, which is 5 - 4 = 1. The height of the rectangle is the difference in y-coordinates between y = 6x and the x-axis for x = 4 and x = 5. The y-coordinate of the line y = 6x for x = 5 is 6(5) = 30, and for x = 4 is 6(4) = 24. Therefore, the height of the rectangle is 30 - 0 = 30. Thus, the area of the rectangle is length * height = 1 * 30 = 30.
Finally, to get the total area of the bounded region, we add the areas of the triangle and the rectangle together: 12 + 30 = 42.
Therefore, the total area of the region described is 42 square units.