Question

Determine whether the integral is convergent or divergent: ∫₀¹ ∫₂⁻⁹x dx −[infinity]. Convergent Divergent. If it is convergent, evaluate it. (If the quantity diverges, enter diverges.)

Answer 1

The given integral is evaluated to be 1/6, indicating that it is convergent.

Step-by-step explanation:

To determine whether the given integral is convergent or divergent, we need to evaluate it.

The given integral is:

∫01 ∫-92 x dx

We first evaluate the inner integral:

∫-92 x dx = [ x2/2 ] (evaluated from -9 through 2)

Substituting the result into the outer integral:

∫01 [ x2/2 ] dx

= 1/2 * ∫01 x2 dx

Integrating this, we get:

= 1/2 * [ x3/3 ](evaluated from 0 through 1)

Substituting the values, we have:

= 1/2 * (13/3 - 03/3) = 1/6

Therefore, the integral is convergent and its value is 1/6.