Question

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations. z = xy, z = 0, y = x, x =1, first octant.

Answer 1

Answer:

1/8

Step-by-step explanation:

Given:

z = xy

z = 0

y = x

x =1

To find:

volume of the solid bounded by the graphs of the equations

Solution:

Compute integral of volume in the first octant:

$Volume = V = \int\limits^1_0\int\limits^x_0 {z} \, dydx$

$\int\limits^1_0\int\limits^x_0 {z} \, dydx = \int\limits^1_0\int\limits^x_0 {xy} \, dydx$

$= \int\limits^1_0x\int\limits^x_0 {y} \, dydx$

= $\int\limits^1_0$ x y²/2 |ˣ₀  dx

= 1/2  $\int\limits^1_0$ x y² |ˣ₀  dx

= 1/2  $\int\limits^1_0$ x (x²-0²) dx

= 1/2  $\int\limits^1_0$ x³dx

=  $\frac{1}{2} \frac{x^{3+1} }{3+1}$ |¹₀

= (1/2) (x⁴/4) |¹₀

= 1/8 x⁴ |¹₀

= 1/8 (1⁴ - 0⁴)

= 1/8 (1)

V = 1/8