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How to calculate the triple integral of a volume bounded by a rotating paraboloid and a plane perpendicular to the Z-axis in spherical coordinates
Let Ω be a bounded closed region bounded by surfaces z = 2-x2-y2 and z = x2 + Y2, and find the volume of Ω
Because the intersection line of Z = 2-2-x2-y2 and z = x2 + Y2 is x2 + y2 = 1, so the projection area of Ω on the xoy surface is D: the projection area of D: D: x 2 + Y2 ≤ 1 65422;, the volume of & nbsp; v = ∭-x2-x2-y2-y2 and z = z = 2 + y2 = 1, so the projection area of Ω on the xoy surface is D: D: D: D: the projection region of D: D: D: D: D: the projection area of D: D: D: D: x 2 + x 2 + Y2 ≤ 2 ≤ 1 65x2 + Y2 ≤ 1 \\\\\\\\\\\\\\\\\= π
The oblique projection of a circle is an ellipse, the projection of an ellipse or an ellipse?
How can I prove mathematically that the oblique projection of a circle on a plane is an ellipse? And the projection of an ellipse on a plane is still an ellipse (it may degenerate into a straight line, etc.). It seems right intuitively, but I don't quite understand the truth. Thank you
For example, the figure after the projection of an ellipse on a plane must still conform to the elliptic equation? I don't know how to prove it mathematically
Hope to give convincing and intuitive proof
thank you!
To: asker W - Ju Ren
It is common sense to draw a straight line from the center of the circle, and then stretch or compress the circle along the two sides of the line to form an ellipse
To: Manager SAMLIN
How can I prove it? I want to find a way to convince my intuition. Projection is not an ordinary coordinate transformation, but also reduced to one dimension. Although I have seen graphics, I don't know how to prove it
Let's say that a circle or an ellipse is a shape composed of numerous short line segments. When the number of line segments is infinite, the circle is perfect. Let's talk about the problem of projection of a regular circle. Let's make a right triangle, and take the line segment a of a regular circle as the hypotenuse of the triangle. Then the length of projection is b = cos (x) a, and X is the angle of projection
When we try to determine the value of K, the intersection of the elliptic paraboloid x ^ + y ^ 2 = 2Z and the plane x = KZ is a circle, and the radius of the circle is calculated
The elliptic paraboloid is x ^ 2 + y ^ 2 = 2Z
The normal vector of plane 2x + 2y-z = 18 is n = (2,2, - 1) because the center of the sphere is (0,0,0), and the line connecting with the center of the formed circle is perpendicular to the plane of the circle, that is, (2,2, - 1) is the radius of the center of the circle r = quadratic
How to change the sine function y = SiNx to get the function y = 1 / 2Sin (3x - π / 3)
First move π / 3 units to the right, then shorten the abscissa to 1 / 3 of the original, and then shorten the ordinate to 1 / 2 of the original
Or we can shorten the abscissa to 1 / 3 of the original, move π / 9 units to the right, and shorten the ordinate to 1 / 2 of the original
If f (x) = 2Sin & sup2; (3x + π / 4), the minimum positive period is?
f(x)=2sin²（3x+π/4）=1-cos2（3x+π/4）=1-cos（6x+π/6）
The formula for calculating the minimum positive period is the coefficient before 2 π / x, so it is π / 3
Given the function f (x) = 2Sin (1 / 3x-30 °), X belongs to R, (1) find the value of F (5 π / 4) (2) let α, β ∈ [0, π / 2], f (3 α + π / 2) = 10 / 13, f (3 β + 2 π) = 6 / 5, find the value of Tan α · Tan β
f(x)=2sin(x/3-30°)=2sin(x/3-π/6)(1)f(5π/4)=2sin(5π/12-π/6)=2sinπ/4=√2.(2)f(3α+π/2)=2sin(α+π/6-π/6)=2sinα=10/13、sinα=5/13、cosα=12/13、tanα=5/12.f(3β+π/2)=2sin(β+π/6-π/6)=2sinβ=6...
It is known that the function f (x) = (Rx ^ 2 + 2) / (s-3x) is odd, and f (2) = - 5 / 3. Find the analytic expression of function f (x)
The function f (x) = (Rx ^ 2 + 2) / (s-3x) is odd, f (- x) = - f (x),
(rx^2+2)/(s+3x)=- (rx^2+2)/(s-3x)
s+3x=- s+3x,s=0.
And f (2) = - 5 / 3, (4R + 2) / (- 6) = - 5 / 3, r = 2
The analytic expression of function f (x) is f (x) = (2x ^ 2 + 2) / (- 3x)
Given the function f (x) = sin (x + π / 6) + 2Sin Λ 2x / 2, find the maximum value of F (x)
f(x)=sin(x＋π/6)＋2sin∧2 x/2
=sin(x＋π/6)＋1-cosx
=sinxcos(π/6)+cosxsin(π/6)+1-2cosxsin(π/6)
=sinxcos(π/6)-cosxsin(π/6)+1
=sin(x-π/6)+1
-2≤f(x)≤2
The maximum value of F (x) is 2
It is known that in △ ABC, Sina + cosa = 1 / 5, which is 3Q
(1) Find the value of sinacosa. (2) judge whether △ ABC is an acute triangle or an obtuse triangle. (3) find the value of Tana
(1) ∵ Sina + cosa = 1 / 5 ℅ (Sina + COSA) = 1 / 25 〉 1 + 2sinacosa = 1 / 25 〉 sinacosa = - 12 / 25 (2) ∵ sinacosa = - 12 / 25 ＜ 0 〉 Cosa ＜ 0 (the value range of triangle inner angle a is 0 ＜ a ＜ 180) ∵ is an obtuse triangle (3) Sina = 1 / 5-cosa 〉 sinacosa = (1 / 5-cosa) cos
RELATED INFORMATIONS
- 1. If the center of the ellipse is the origin, the focus is on the x-axis, the point P is a point on the ellipse, and the projection of P on the x-axis is exactly the origin of the ellipse The center of an ellipse is at the origin, the focus is on the x-axis, P is a point on the ellipse, the projection of P on the x-axis is just the left focus of the ellipse, the line between P and center O is parallel to the line between the right vertex and the upper vertex, and the distance between the left focus and the left vertex is equal to √ 10 - √ 5. Try to find the eccentricity and equation of the ellipse
- 2. Normal equation of curve y = x ^ 3-3x at point (2,2) If the problem, solve!
- 3. Is the hyperbolic paraboloid obtained by y = ax ^ 2 rotating around the X axis? But how to deduce the hyperbolic paraboloid equation? If y = ax ^ 2 rotates around the X axis, we should get √ y ^ 2 + Z ^ 2 = ax ^ 2. Can this equation change to hyperbolic parabolic equation?
- 4. Find the volume of solid surrounded by z = x ^ 2 + y ^ 2 and z = 2-radical (x ^ 2 + y ^ 2)
- 5. Calculate the surface integral ∫ ∫ zdydz, where ∑ is the lower side of the cone z = x ^ 2 + y ^ 2 between the plane z = 0 and z = 3
- 6. Find the volume of solid surrounded by x ^ 2 + y ^ 2 under the sign of cone z = root, x ^ 2 + y ^ 2 = 1 on cylinder and z = 0 on plane
- 7. This is an example in the book: finding the volume of the solid surrounded by a rotating paraboloid A: X & # 178; + Y & # 178; = R & # 178; and B: X & # 178; + Z & # 178; = R & # 178
- 8. ∫∫ x ^ 2dydz + y ^ 2dzdx + Z ^ 2dxdy, where the surface is outside the upper part of x ^ 2 + y ^ 2 + Z ^ 2 = 1
- 9. We know that x, y and Z are equal difference sequence. We prove that x2 [y + Z], Y2 [x + Z], Z2 [x + y] are equal difference sequence
- 10. X2 + Y2 + Z2 = 1 to find the value range of X + y
- 11. Find the intersection angle of two curves x + y = 0, XY '= 0 on the surface z = XY The two curves should be x + y = 0, xy = 0
- 12. Calculate ∫ ∫ ∑ (x ^ 2 + y ^ 2) ds, where ∑ is the part of paraboloid z = 2-x ^ 2 + y ^ 2 above the xoy plane If ∑ is the entire boundary surface bounded by the cone z = √ (x ^ 2 + y ^ 2) and the plane z = 1, how to find it?
- 13. Surface x2 + Y2 + Z = 9 tangent plane equation at point 124
- 14. The tangent plane equation at point P (1.1.1)
- 15. It is proved that the volume bounded by the tangent plane of any point on the cubic (a > o) of the surface XYZ = A and the three coordinate planes is a certain number The answer is: surface XYZ = A & # 179; the normal direction of (x0, Y0, Z0) is {y0z0, z0x0, x0y0} The tangent plane is: y0z0 (x-x0) + z0x0 (y-y0) + x0y0 (z-z0) = 0 Its intercept on three coordinate axes are 3x0, 3y0 and 3z0 The volume of the tetrahedron bounded by the tangent plane and three coordinate planes is: 27x0y0z0 / 6 = 9A & # / 2 My question is, how do you calculate the intercept above!
- 16. The geometric meaning of double integral is volume, but double integral can produce negative number,
- 17. Double integral of basic high number 1.∫∫D（x²-y²)dxdy ,0
- 18. Calculating volume by double integral V is the sphere x ^ 2 + y ^ 2 + Z ^ 2
- 19. Party A and Party B want to jointly contract a project. Party A will complete the project within 30 days and Party B will complete the project within 20 days. The contract stipulates that the project will be completed within 15 days. Otherwise, a fine of 1000 yuan will be imposed for each day. Party A and Party B sign the contract after discussion. (1) under normal circumstances, can party A and Party B perform the contract? Why? (2) Now they have cooperated in 75% of the project. Because there is something urgent elsewhere, one person must be transferred. Who is more suitable? Why?
- 20. Using spherical coordinates to calculate triple integral I = ∫∫∫ Z ^ 2dV, where the graph is composed of x ^ 2 + y ^ 2 + Z ^ 2