If the curve V is the square of the surface y = x, y = x, x + y + Z = 2, and z = 0 is the defined area, then fffdxdydz =? Calculus

If the curve V is the square of the surface y = x, y = x, x + y + Z = 2, and z = 0 is the defined area, then fffdxdydz =? Calculus

∫∫∫dxdydz=∫(0,1)dx∫(x^2,x)dy∫(0,2-x-y)dz
=∫(0,1)dx∫(x^2,x)(2-x-y)dy=∫(0,1)(2x-(7/2)x^2+x^3+(1/2)x^4)dx=1-7/6+1/4+1/10=11/60

Find the plane equation which passes through the line 2x + y = 0,4x + 2Y + 3Z = 6 and is tangent to the sphere x ^ 2 + y ^ 2 + Z ^ 2 = 4

Simultaneous 2x + y = 0,4x + 2Y + 3Z = 6
The result is: z = 2
So: we know that the line is on the plane z = 2
And: the center of the sphere x ^ 2 + y ^ 2 + Z ^ 2 = 4 is at the origin, and the radius is 2
So: z = 2 is the section of the ball
So, the plane equation is: z = 2

The tangent plane equation of X + y + Z + 14 at points (1,2,3) is? The normal equation is?

Square of sphere x + square of Y + square of Z = a the tangent plane equation at points (x0, Y0, Z0) is?
x0 x+y0 y+z0 z=a;
The normal equation is
x-x0=l x0
y-y0=l y0
z-z0=l z0
Eliminate L is

On the problem of plane equation in higher mathematics! 1. Find the plane equation through (x-1) / 2 = (y + 2) / 3 = (Z + 3) / 4 and parallel to the straight line x = y = Z / 2
1. Find the plane equation through (x-1) / 2 = (y + 2) / 3 = (Z + 3) / 4 and parallel to the straight line x = y = Z / 2
2. Find the linear equation on the plane x + y + Z = 0 and intersect with two straight lines L1: x + Y-1 = 0, X-Y + Z + 1 = 0; L2: 2x-y + Z-1 = 0, x + Y-Z + 1 = 0

Question 1: let (x-1) / 2 = (y + 2) / 3 = (Z + 3) / 4 be L1, x = y = Z / 2 be L2, the parallel vector of L1 is (2,3,4) temporarily denoted as u, the parallel vector of L2 is (1,1,2) temporarily denoted as V, let w = u × V, then w = (2,0, - 1)

As shown in the figure, find the section equation of a point on the elliptic plane, which is parallel to the given plane

Let the tangent point be (a, B, c), then the equation of tangent plane is ax + 2by + CZ = 1. Since the tangent plane is parallel to the known plane, a / 1 = 2B / (- 1) = C / 2, and a ^ 2 + 2B ^ 2 + C ^ 2 = 1, the solution is a = - √ 22 / 11, B = √ 22 / 22, C = - 2 √ 22 / 11 or a = √ 22 / 11, B = - √ 22 / 22, C = 2 √ 22 / 11

Find the volume of the solid surrounded by the sphere x ^ 2 + y ^ 2 + Z ^ 2 = 2 and the paraboloid z = x ^ 2 + y ^ 2

Volume = ∫ d [√ (2-x & # 178; - Y & # 178;) - (X & # 178; + Y & # 178;)] DXDY
Do it in polar coordinates

The triple integral ∭ & nbsp; Ω (x2 + Y2 + Z2) DV is calculated, where Ω is a closed sphere surrounded by x2 + Y2 + Z2 = 1

According to the meaning of the title, Ω = {(R, φ, θ) | 0 ≤ θ ≤ 2 π, 0 ≤ φ ≤ π, 0 ≤ R ≤ 1} ∫∫∫ Ω (x2 + Y2 + Z2) DV = ∫ 2 π 0d θ∫ π 0sin φ D φ∫ 10r4dr = 2 π · [− cos φ] π 0 · [15r5] 10 = 4 π 5

There are several ways to solve double integral

One: change to twice integral
There are two coordinate systems: rectangular coordinate system and polar coordinate system
There are two basic types of quadratic integrals in Cartesian coordinates: X-type and Y-type

On the formula of double integral calculation
Today, I watch TV to explain the calculation! Slice volume: v = s (x) *
It says that this is approximately equal! It should be very thin, so it can be regarded as the volume of cylinder! 1. Only the volume of cylinder is the low area multiplied by the high? What is the standard cylinder, can the cube apply that formula? 2. What's more, he cut the surface of the cylinder! It's not the standard volume of cylinder! Thin has nothing to do with it's the standard volume of cylinder!

It seems that you are not generally dizzy. Let me explain briefly. Only the volume formula of the cylinder is the bottom multiplied by the height, just like the parallelogram is the bottom multiplied by the height in the plane. The volume of the vertebral body is 1 / 3 of the bottom multiplied by the height, and the volume of the platform is subtracted by the upper and lower vertebral bodies

Help double integral calculation!
∫∫ (3x + 2Y) DXDY, where D is a closed region bounded by two coordinate axes and a straight line x + y = 2
D

Idea: integration by parts
First, we integrate (3x + 2Y) with respect to y from 0 to 2-x, and then with respect to x from 0 to 2
Original integral = 6 * x * (2-x) + 2 * (2-x) ^ 2