Calculating volume by double integral V is the sphere x ^ 2 + y ^ 2 + Z ^ 2

Calculating volume by double integral V is the sphere x ^ 2 + y ^ 2 + Z ^ 2

First, we calculate the intersection line of two spheres. It is easy to get that the intersection line is Z = R / 2, and the plane z = R / 2 divides the common part into two parts. The two parts are symmetrical, so we only need the upper part, and then double it. Substituting Z = R / 2 into the spherical equation, we get: X & # 178; + Y & # 178; = 3R & # 178; / 4. Therefore, this problem is transformed into calculating the spherical X & # 178; + Y &

Double integral calculation volume
The plane figure D is surrounded by curve, line and axis. (1) calculate the area of the plane figure; (2) calculate the volume of the revolving body formed by the plane figure rotating around the axis

Let me talk about the general situation
Let a plane figure d be bounded by a curve y = f (x), a straight line x = a, x = B, b > A and X axis
Then: 1. The area of plane figure s = ∫ [a, b] f (x) DX
2. The volume of the rotator formed by the rotation of the plane figure around the axis:
Using the differential element method, if any point x is taken in the interval [a, b], then s (x) = π f (x) ^ 2
So: v = ∫ [a, b] π f (x) ^ 2DX

Comparison of double integral

The previous integral is larger, and the analysis process is shown in the figure. The economic mathematics team will help you to solve it, please adopt it in time. Thank you!

On the volume of definite integral
Why use definite integral to calculate volume? What's the basic principle
And what does dy mean in definite integral?

The definite integral volume is based on the idea of micro element method, which divides the solid into many thin slices with several parallel sections, just like you cut a radish with a knife. Note that the two ends of the radish must have been cut off to form a section sandwiched in two parallel sections, and each slice of radish is approximately regarded as a column, and the bottom area is

What is the calculation formula of area volume arc length in the application of definite integral

Use the property and definition of definite integral to express the area of the plane area enclosed by the following curve
Y = X-2, x = Y2 (the square of Y)

First, find the intersection point of two curves and establish two equations
y=x-2
x=y²
The solution is X1 = 1, Y1 = - 1
x2=4,y2=2
The intersection points are (1, - 1) and (4,2)
Between the two intersections, the curve X = y & sup2; is above y = X-2
The area of the plane area enclosed by the curve
S=∫ (y+2)dy-∫ (y²)dy
=(y²/2+2y-y³/3)|
=2+4-8/3-1/2+2-1/3
=9/2

The application of definite integral to calculate area
Find the area of the common part of P = 3cost and P = 1 + cost!
The parameter only has the formula to calculate the arc length~
How to solve this problem?
1F the answer is wrong 5 / 4 π
In addition, how to find the range of T?

The solutions P = 3cost and P = 1 + cost give t = ± π / 3
The area of the common part = ∫ (- π / 3, π / 3) (3cost-1-cost) DT
=∫（-π/3,π/3）（2cost-1）dt
=（2sint-t）|（-π/3,π/3）
=2√3-2π/3.

Double integral for triangle area
Given three points in the coordinate system, a (0, - 1) B (1,1) C (3,1) use double integral to find the area of the triangle

It can be counted, but the integral number is really hard to play. S = ∫∫ DXDY
x. The interval of Y integral is:
y/2+1/2 < x < 3y/2+3/2
-1

Confused high number, double integral for volume, triple integral is also for volume, what's the difference between them? What's the use of triple integral?
Talk about the image

Let's put it this way
The definite integral can find the area, and the double integral can also find the area,
I understand that
The truth is the same
But integral can't be understood as area or volume
Seeking area or volume is only a geometric application of integral
For triple integral, only if integrand = 1 is volume
For general integrand function, for example, it can be understood as finding the mass of space object with non-uniform density

Who can tell me vividly what double triple integral is?
A bunch of things, including the integral region The single integral is the area of the lines of different lengths, double and triple

OK, let's use the graph to illustrate (the quadratic curve in the rectangular plane coordinate system, above the x-axis) how to integrate the quadratic function f (x) on the x-axis, that is, the area enclosed by it and the x-axis
For indefinite integral, its range on x-axis is not limited. It represents a dynamic range. Specifically, it is a function
The definite integral is to limit a range, such as (- 8,6), so that we can calculate the area of F (x), x = - 8, x = 6, and X axis
Double integral, double integral is a function of one person's two variables, such as Z = f (x, y). It is a three-dimensional figure of space, it is a projection in the X, y plane, and the volume of space is double integral. This is a bit abstract, not very easy to say. If you really want it, I can tell you in detail
Triple integral can only be visualized in four-dimensional space, so it can only be imagined with mathematical thinking. It is analogized with double integral and integral. Only when you understand integral, double and triple are not afraid
These can be applied to all aspects. For example, you can calculate the volume of an irregular object. Many aspects can be converted into the area of calculus. This is its great advantage. This area and volume is a concept of abstraction, and more multiple integrals will have more meaning