Question

Evaluate the integral using the following values.

⁸∫ ₂ x³ dx = 1,020, ⁸∫ ₂ x dx = 30, ⁸∫ ₂ dx = 6.
⁸∫ ₂ (x-17) dx.

Answer 1

The integral Ⅰ∫ ₂ (x-17) dx is evaluated by separating it into the integral of x minus 17 times the integral of 1, both from 2 to 8, and summing the provided values of these integrals to get -72.

Step-by-step explanation:

To evaluate the integral of (x-17) with the given values from 2 to 8, we can separate the integral into two parts using the properties of integrals:

• The integral of x from 2 to 8: ⁸∫ ₂ x dx = 30.
• The integral of -17 from 2 to 8: ⁸∫ ₂ -17 dx = -17 × ⁸∫ ₂ dx = -17 × 6 = -102.

Adding the two results together, we get:

30 - 102 = -72.

Therefore, the value of the integral ⁸∫ ₂ (x-17) dx is -72.

Answer 2

To evaluate the integral ⁸∫ ₂ (x-17) dx, we subtract 17 times the integral from 2 to 8 of dx (which is 6) from the integral from 2 to 8 of x dx (which is 30). This results in 30 - 102, giving us a final answer of -72.

Step-by-step explanation:

To evaluate the integral ⁸∫ ₂ (x-17) dx, we can use the properties of integrals to simplify this into simpler parts, for which we already know the values. We can express the integral as the difference between the integral of x and the integral of 17 from 2 to 8.

Using the given information:

1. ⁸∫ ₂ x dx = 30
2. ⁸∫ ₂ dx = 6

We can determine:

⁸∫ ₂ (x-17) dx = ⁸∫ ₂ x dx - 17 × (⁸∫ ₂ dx)

Replace the known values:

⁸∫ ₂ (x-17) dx = 30 - 17 × 6

⁸∫ ₂ (x-17) dx = 30 - 102

⁸∫ ₂ (x-17) dx = -72

Therefore, the value of the integral from 2 to 8 of (x-17) is -72.