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Problems Encountered by Student

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PROJECT
IN
INTEGRAL CALCULUS

SUBMITTED BY: JAYCER D TIMBALOPEZ BSEE III-A
SUBMITTED TO: ENGR. ASER DINO

FIRST MOMENT OF A PLANE AREA

1) Find the Mx for the area bounded by y2=4x and x2=4y.
INTERSECTION:
(y24)2=4y y4=64y yy3-64=0 y=0:y=4 SOLUTION:
Mx=04y(2y-y24)dy
Mx=04( 2y32-y33)dy
Mx=4y53-y41604
Mx=495

2) Find the moment of area for the first quadrant bounded by y2=4x, the x-axis and x=4with respect to x=4.
SOLUTION:
MA=044-x2xdx =04(8x-2x32) dx =16x323-4x52504
MA=25615

CENTROID OF A PLANE AREA
1) Find the centroid of the area enclosed by x+2y=6, x=0 and y=0.
SOLVE FOR THE AREA:
A=066-x2dx
=3x-x2406
A=9 sq.units
SOLVE FOR My: my=06xcdA =06x6-x2dx =063x-x22dx = 3x22-x3606 my=18 SOLVE FOR Mx: mx=03y6-2ydy =036y-2y2dy =3y2-2y3303 mx=9 SOLVE FOR x: SOLVE FOR y: x=myA y=myA =189 = 99 x=2 y=1
2) Find the centroid enclosed by r=3sinθ, θ=π2 (first quadrant)
SOLVE FOR THE AREA:
A=120π2r2dθ
=120π2sin2θdθ
=920π21-cos2θ2dθ
=92θ-12sin2θ0π2
A=9π8
My=130π227sin3θcosθdθ
=9sin4θ40π2
My=94

Mx=130π227sin4θdθ
=90π2sin4θdθ
=93142*π2
Mx=27π16
Solve for x: x=MyA=94*89π=2π Solve for y: y=MxA=27π16*89π=32 CENTROID OF A SOLID REVOLUTION

-Find the centroid of the solid generated by revolving about the indicated axis the area bounded by the given curves.

1)y2=x, y=3, x=0:about y axis
V=π03x2dy
=π03y4dy
=πy5503
V=2435π
Mxz=π03yy4dy
=π03y5dy
=y6603
mxz=243π2
SOLVE FOR y: y=MxzV =243π2*5243π y=52 C:0,52,0

2)x2=4y, y2=4x:about y-axis
SOLVE FOR VOLUME:
V=π042y2-y242dy
=π044y-y416dy
=2y2-y58004
V=96π5

SOLVE FORMxz:
Mxz=π04y4y-y416dy
Mxz=π044y2-y516dy
Mxz=4y33-y69604
Mxz=128π3
SOLVE FORy: y=MxzV y=128π3*596π y=209 C:0,209,0

MOMENT OF INERTIA OF A PLANE AREA

-Find the moment of inertia of the area bounded by the given curves with respect to the specified axis.
1)2x+y=6, x=0,y=0:with respect to x-axis
Ix=06y2dA
=06y26-y2dy
=12066y2-y3dy
=122y3-y4406
Ix=54

2) y=4x2, y=4x:with respect to y-axis
Iy=01x2dA
=01x24x-42dx
=014x2-4x4dx
=x4-4x5501
Iy=15

MOMENT OF INERTIA OF A SOLID REVOLUTION
-If the area bounded by the given curves is revolved about the specified axis, find the moment of inertia of the solid generated with respect to the axis of revolution.
1) 2x+3y=6, x=0;y=0:about x-axis
Ix=02y2dV
Ix=02y22πy6-3y2dy
Ix=π02y36-3ydy
Ix=π026y3-3y4dy
Ix=π3y42-3y5502
Ix=245π

2)xy=4, y=x, y=1, about y=o
Ix=12y2dV
Ix=12y22πy4y-ydy
Ix=2π12y34y-ydy
Ix=2π124y2-y4dy
Ix=2π4y33-y5512
Ix=9415π

THE DOUBLE INTEGRAL

1)01x4-3xdydx
=01yx4-3xdx
=014-3x-xdx
=-23*4-3x323-2x32301
=-89

2) 0113xydxdy
= 01x2y213dy
= 019y-y2dy
=014ydy
=2y201
=2

PLANE AREA BY DOUBLE INTEGRATION

1) Find the area bounded by y=x3 and y=x (first quadrant) using double integration.
A=01x3xdydx
A=01yx3xdx
A=01(x-x3)dx
A=x22-x4401
A=12-14
A=01x3xdydx
A=14sq.units

2)The area enclosed by r=4cosθ
A=0π204cosθrdrdθ
A=0π2r2204cosθ
A=0π216cos2θdθ
A=80π21+cos2θdθ
A=8θ+12sin2θ0π2
A=8π2
A=4π

VOLUME OF SOLID REVOLUTION (BY DOUBLE INTEGRATION)

1) Area bounded by y=x3, x=2, y=0;about the y-axis
V=2π08y132xdxdy
V=2π08x22y132
V=π084-y23dy
V=π4y-3y53508
V=645π

2) Find by double integration the volume generated by revolving about the polar axis the upper half of the circle r=6cosθ.
V=2π0π206cosθr2sinθdrdθ
V=2π0π2r33sinθ06cosθdθ
V=144π0π2cos3θsinθdθ
V=144π-cos4θ40π2
V=36π cubic units

CENTROID OF A PLANE AREA (BY DOUBLE INTEGRATION)
1) Find the centroid of the area enclosed by x+2y=6, x=0 and y=0.
A=0606-x2dydx
A=066-x2dx
A=3x-x2406
A=9 sq.units

My=0606-y2dydx
My=06x6-x2dx
My=063x-x22dx
My=3x22-x3606
My=18

Mx=060x-62ydydx x=MyA=189=2 units Mx=06y2206-x2dx
Mx=06x2-12x+36dx y=MxA=99=1 unit
Mx=18x33-6x2+36x06
Mx=9

2)r=3sinθ, θ=π2
A=0π203sinθrdrdθ
A=0π2r2203sinθdθ
A=0π29sin2θ2dθ
A=940π21-cos2θdθ
A=94θ-12sin2θ0π2
A=9π8 SOLVE FOR My: SOLVE FOR Mx: My=0π203sinθr2cosθdrdθ Mx=0π203sinθr2sinθdrdθ My=0π2r3303sinθdθ Mx=0π2r3303sinθ My=0π29sin3θcosθdθ Mx=0π29sin4θdθ My=9sin4θ40π2 Mx=9*4-1(4-2)(4-2)*π2 My=94 Mx=27π16

SOLVE FOR CENTROID: x=MyA=94*89π=2π y=MxA=27π8*89π=32

MOMENT OF INERTIA OF A PLANE AREA AND POLAR MOMENT (by DOUBLE INTEGRATION)

*Find Ix,Iy,and I0 of the area bounded by the given curves.

1) 2x+y=6, x=0 and y=0
SOLVE FOR Ix:
Ix=0606-y2y2dxdy
Ix=06y2x06-y2
Ix=06y26-y2dy
Ix=1206(6y2-y3)dy
Ix=122y3-y4406
Ix=54

SOLVE FOR Iy:
Iy=0606-y2x2dxdy
Iy=06x3306-y2
Iy=124066-y 3
Iy=06216-108y+18y2-y3dy
Iy=216y-54y2+6y3-y4406
Iy=272

SOLVE FOR I0:
I0=0606-y2x2+y2dxdy
I0=06x33+y2x06-y2dy
I0=06216-108y+18y2-y324+6y2-y32
I0=12406216-108y+90y2-13y3
I0=124216y-50y2+30y3-13y4406
I0=1352

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