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Use geometry to determine the value of ∫⁶₀ (x+1)dx

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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.

User Evan Sharp
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