Final Answer:
The area of the region under the given curve from 1 to 5 is 49 units.
Step-by-step explanation:
The given equation is y = x² - 7x + 6x - x² which can be simplified to y = 2x - 7x + 6. To find the area under the curve, we will use the definite integral method. The definite integral of a function from a to b is the area between the curve of the function and the x-axis from x = a to x = b. The definite integral of the given equation is ∫2x-7x+6 dx.
To solve the definite integral, we will use the method of integration. The integration of 2x-7x+6 is x² - 7x + 6 + c, where c is a constant. Substituting the lower limit of the definite integral, i.e., x = 1, we get 1 - 7 + 6 + c = c + 0. Similarly, substituting the upper limit of the definite integral, i.e., x = 5, we get 25 - 35 + 6 + c = c - 4. Adding 0 and -4, we get c = -4. Hence, the integration of the given equation is x² - 7x + 6 - 4.
To find the area of the region under the given curve from 1 to 5, we have to subtract the definite integral from the lower limit to the upper limit. Hence, the area of the region under the given curve from 1 to 5 is ∫2x-7x+6 dx from 1 to 5 = (5² - 7x + 6 - 4) - (1² - 7x + 6 - 4) = 49 units.