Question

The volume determined from \[ \int \int \int_V 8 x y z \, dV \text{for} V = [2, 3] \times [1, 2] \times [0, 1] \] will be (in integer) \(\underline{\hspace{1cm}}\).

Hint:

When calculating a triple integral, perform the integrations step by step, starting with the innermost integral.

Answer 1

Correct Answer: 15

We calculate the triple integral: \[ \int_2^3 \int_1^2 \int_0^1 8 x y z \, dz \, dy \, dx \] First, integrate with respect to \( z \): \[ \int_0^1 8 x y z \, dz = 8 x y \left[ \frac{z^2}{2} \right]_0^1 = 4 x y \] Next, integrate with respect to \( y \): \[ \int_1^2 4 x y \, dy = 4 x \left[ \frac{y^2}{2} \right]_1^2 = 4 x \cdot \frac{3}{2} = 6 x \] Finally, integrate with respect to \( x \): \[ \int_2^3 6 x \, dx = 6 \left[ \frac{x^2}{2} \right]_2^3 = 6 \cdot \left( \frac{9}{2} - \frac{4}{2} \right) = 6 \cdot \frac{5}{2} = 15 \] Thus, the volume is \( \boxed{15} \).