To calculate the total area of the region bounded by the line y = 39x, the x-axis, and the lines x = 10 and x = 16, we need to find the area between the curve and the x-axis within the given x-range.
First, let's find the points of intersection between the line y = 39x and the lines x = 10 and x = 16.
For x = 10:
y = 39 * 10 = 390
For x = 16:
y = 39 * 16 = 624
So, the points of intersection are (10, 390) and (16, 624).
To find the area between the curve and the x-axis, we can integrate the function y = 39x within the given x-range.
The integral of y = 39x with respect to x is:
∫(39x) dx = (39/2)x^2 + C
To find the area, we need to evaluate the definite integral between x = 10 and x = 16:
Area = ∫[10, 16] (39x) dx
= [(39/2)x^2] [10, 16]
= [(39/2)(16)^2 - (39/2)(10)^2]
= [(39/2)(256) - (39/2)(100)]
= (9984 - 1950)
= 8034 square units
Therefore, the total area of the region bounded by the line y = 39x, the x-axis, and the lines x = 10 and x = 16 is 8034 square units.