DOUBLE INTEGRALS : INTRODUCTION #double #integrals #introduction
DOUBLE INTEGRALS
The line integrals are often used to find the lengths of the curves i.e if we talk about a circle the line integral is the circumference of the circle or semi circle or quarter circle etc. If we ask about the length of the curve on the circle which is not a complete circle or semi or quarte circle , we chose the part of the circumference of the circle then we need some procedure or theory to find the length of the curve which are called the line integrals.
This process of line integration is applicable not only for the circles but also for any one or two or three dimensional regular and irregular shapes .
So what about the surface area of circular cake in a rounded plate or cool drink bottle looks like a cylinder or football like a sphere etc. In mathematics there are certain formulae for the above mentioned shapes. But what about the surface area of the irregular shapes like a part of saree which is teared or a peace of a broken jar or wave of a water layer occured due to dropping of a stone in a still water etc. For this the concept of double integral is introduced.
Here in this page we see the double integral of a bounded function over a rectangle in R2
Double integrals are a form of integration used to calculate values over a two dimensional region, extending the concept of single integrals to functions of two variables. They are often used to find areas, volumes or other quantities in a space where one variable is a function of another.
Let R be the region of integration in the XY-plane and f(x,y) be a bounded function on the region R.
∴ The double integral of f is denoted by ∬f(x,y)dxdy over the region R.
Here
* R is region of integration.
* a, b are the limits of integration for x.
* c, d are the limits of integration for y.
* dx, dy indicates the order of integration.
Double integrals are a powerful tool for the concept of integration to two dimensions, the calculation of various quantities over areas and surfaces.
The concept of a double integral of f(x,y) over a rectangle in R2
2. 2. Evaluate ∬xy (x2+y2)dx dy over [0,a;0,b].
3. 3. Evaluate ∬ yexy over [0,a;0,b].
4. 4. Evaluate ∬(x-y)/(x+y) dxdy over [0,1;0,1].
7. 7. Evaluate ∬y2/(1+x2) dxdy over [-1,1;0,2].
8. 8. Integrate ∬f(x,y) =y cosxy over the rectangle 0≤x≤π , 0≤y≤1
9. 9. Evaluate the double integral of f(x,y) =xy(x2+y2) over the rectangle R = [0,1;0,1].
10. Evaluate ∬ f(x,y) dxdy where f(x,y) = √x + y-3x2y and R= [0,1;1,3].
12. Evaluate ∬ xy dxdy over the region E which is bounded by xy=1, y=0, y=x, x=2.
13. Evaluate ∬√(4x2-y2) dxdy over E where E is the triangle bounded by the lines y=0, y=x, x=1.
14. Evaluate ∬ xy (x+y) dxdy over E where E is the region bounded by y=x2 and y=x.
15. Evaluate ∬ f(x,y) dxdy over E where f(x,y)= x2+y2 and E={ (x,y)/ y=x2 , x=2,y=1}
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