Final answer:
The value of the integral ∫⁶₀ (x+1)dx is 24, which can be determined using the geometric interpretation of an integral as the area under a curve; in this case, it corresponds to the area of a trapezoid.
Step-by-step explanation:
To determine the value of ∫⁶₀ (x+1)dx, we can use geometry by interpreting the integral as the area under the curve f(x) = x + 1 from x = 0 to x = 6. This can be visualized as a trapezoid with bases of lengths 1 and 7 (since f(0) = 1 and f(6) = 7) and a height of 6. Calculating the area of a trapezoid is done with the formula A = ½ * (b1 + b2) * h, where b1 and b2 are the lengths of the two bases, and h is the height.
Substituting the values we have:
- b1 = f(0) = 1
- b2 = f(6) = 7
- h = 6
Thus, using the trapezoidal area formula, we get:
A = ½ * (1 + 7) * 6 = ½ * 8 * 6 = 24.
Therefore, the value of the definite integral ∫⁶₀ (x+1)dx is 24.