5 Tasks On Double Integrals To Be Done In Mathcad essay

Kodune kontrolltöö nr.3

1. Vahetage kahekordses integraalis

∫ ∫ 2

4− 2

0

2

0

integreerimisjärjekord, esitage integreeritav piirkond joonisel ning leidke integraali

väärtus.

2. Leida pinna D pindala, kui = {( , ) | ≥ 1 − 2, ≤ 4 − 2, ≥ 0, ≥ 0}.

3. Leida keha ruumala, mis ülalt on tõkestatud pinnaga = 2 + 2 ning põhjaks on xy-

tasandil joonte = 2 ja = 3 poolt piiratud pind.

4. Leida integraali

∫ ∫ ln( 2 + 2 + 1)

√1− 2

−√1− 2

1

−1

väärtus minnes üle polaarkoordinaatidele.

5. Antud on jooned = 2 , = 4 ja = 4.

a) Esitada joonte graafikud.

b) Leida joonte lõikepunktid.

c) Leida joonte poolt piiratud pinnatüki pindala.

d) Leida pinnatüki mass, kui integreeritava pinnatüki pindtihedus on

( , ) = 4 2 + 1.

e) Leida pinnatüki massikeskme koordinaadid ning esitada massikese joonisel.

f) Leida inertsmomendid , ja 0.

1. In the given double integral

a. Change the order of intergration (from dydx to dxdy) – with steps shown!

b. Show the integration area on xy plot

c. Find the value of integral

2. Find the area of surface D which is surrounded by given constraints. Show the area on xy plot.

3. Find the volume of a body, where the upper surface is constrained by z and bottom surface is

surrounded by two given lines x.

4. Find the value of double integral via switching to polar coordinates. Show it on plot.

5. 3nr y lines are given

a. Show the lines on xy plot

b. Find the cutting points of lines

c. Find the area of surface surrounded by these lines

d. Find the mass of the surface when its density is ρ(x,y)

e. Find the coordinates of the mass centre point of the surface and indicate it on plot

f. Find the listed moments of inertia

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