How do you find the volume of a solid in the first octant?

Find the volume of the solid in the first octant bounded by the cylinder z = 16 – x2 and the plane y = 5? Solution: The volume of the solid can be found using the triple integral. Therefore, the volume of the solid is 32/3.

How do you find the 1st octant?

The first octant is a 3 – D Euclidean space in which all three variables namely x , y x, y x,y, and z assumes their positive values only. In a 3 – D coordinate system, the first octant is one of the total eight octants divided by the three mutually perpendicular (at a single point called the origin) coordinate planes.

What is the first octant of a plane?

The three planes all intersect in one point, the origin (located at (0,0,0)), and divide 3 space into 8 octants (similar to the 4 quadrants in 2 dimensions). The octant in which all three coordinates are positive is called the first octant.

What is the first octant in cylindrical coordinates?

z3√x2 + y2 + z2dV , where D is the region in the first octant which is bounded by x = 0, y = 0, z = √x2 + y2, and z = √1 − (x2 + y2). Express this integral as an iterated integral in both cylindrical and spherical coordinates.

What is Theta in first octant?

The possible values for theta are any values in the first octant, so we need 0 <= theta <= pi/2. Now in each slice, we want to think about forming a double integral in the remaining variables (r and z).

What is the volume of the solid?

The volume of a solid is the measure of how much space an object takes up. It is measured by the number of unit cubes it takes to fill up the solid. Counting the unit cubes in the solid, we have 30 unit cubes, so the volume is: 2 units 3 units 5 units = 30 cubic units.

How do you find the volume of cylindrical coordinates?

Finding volume for triple integrals in cylindrical coordinates

  1. V = ∫ ∫ ∫ B f ( x , y , z ) d V V=\int\int\int_Bf(x,y,z)\ dV V=∫∫∫B​f(x,y,z) dV.
  2. where B represents the solid cylinder and d V dV dV can be defined in cylindrical coordinates as.
  3. d V = r d z d r d θ dV=r\ dz\ dr\ d\theta dV=r dz dr dθ

What is an octant in math?

octant. / (ˈɒktənt) / noun. maths. any of the eight parts into which the three planes containing the Cartesian coordinate axes divide space.