Final answer:
To find the area of the region under the given curve, we need to calculate the definite integral of the function over the interval from 1 to 6. The area is -185/3 square units.
Step-by-step explanation:
To find the area of the region under the given curve, we need to calculate the definite integral of the function over the interval from 1 to 6. The given function is y = x² - 3x + 7x - x². Let's simplify this expression:
y = 4x - x²
Now, we can integrate this function from 1 to 6:
∫16 (4x - x²) dx
To evaluate this integral, we can use the power rule and the linear rule of integration:
= [(4x²/2) - (x³/3)]16
= [(24/2) - (216/3)] - [(4/2) - (1/3)]
= [12 - 72] - [2 - (1/3)]
= -60 - (5/3)
= -180/3 - 5/3
= -185/3
Therefore, the area of the region under the curve from 1 to 6 is -185/3 square units.