Mathematical Scratchpad

1  Double Integral Example Worksheet

1.0.1  Double Integrals over general regions in x,y coordinates

Sketch regions too  

1. ò₀⁴ò₀^4-x xy dy dx

   Inner: ò₀^4-x xy dy=\allowbreak [ 1/2]xy²| _y=0^4-x=[ 1/2]x( -4+x) ²

   Completion: ò₀⁴[ 1/2]x( -4+x) ²dx=[ 1/2]ò₀⁴( x³-8x²+16x)  dx=\allowbreak [ 32/3]

2. \iint_D( x+y)  dA  where D is the triangle with vertices ( 0,0) ,( 0,2) ,( 1,2)

   ò₀¹ò_y=2x²( x+y) dydx=ò₀²ò₀^[ 1/2]y( x+y) dxdy=[ 5/3].

3. \iint_D48xy  dA   where D is the region bounded by y=x³  and y=Öx

   ò01òy=x3Öx( 48xy)  dy dx=\allowbreakò01òx=y2y[ 1/3]( 48xy) dx dy=\allowbreak 5

Reverse order of integration.  

1. ò₀¹ò_x^2xe^y-x  dy dx=ò_y=0¹ò_x=[ 1/2]y^ye^y-x dx dy+ò₁²ò_[ 1/2]y¹e^y-x dx dy
2. ò₀^2Ö3ò_y²/6^Ö{16-y2}1  dx dy=ò₀²ò₀^Ö{6x}1 dy dx+ò₂⁴ò₀^Ö{16-x2}1 dy dx=\allowbreak [ 2/3]Ö3+[ 8/3]p
3. ò₀⁷ò_x²-6x^x f( x,y)  dy  dx=ò_-9⁰ò_3-Ö{y+9}^3+Ö{y+9}f( x,y) dx dy+ò₀⁷ò_y^3+Ö{y+9}f( x,y)  dx dy
4. ò12òxx3 f( x,y) dy dx+ò28òx8 f( x,y) dy dx=ò18òy[ 1/3]yf( x,y)  dx dy

Find Volume of solid  

1. Tetrahedron in first octant bounded by coordinate planes and z=7-3x-2y.

   ò₀^[ 7/3]ò₀^{-[ 3/2]x+[ 7/2]}(7-3x-2y)  dy dx=ò₀^[ 7/3]ò₀^7-3x( [(7-3x-z)/2])  dz dx=ò₀^[ 7/2]ò₀^7-2y( [(7-2y-z)/3])  dz dy=\allowbreak [ 343/36]

2. Solid inside both the sphere x²+y²+z²=3  and paraboloid 2z=x²+y².

   ò_-Ö2^Ö2ò_-Ö{2-x²}^Ö{2-x2}( Ö{3-x²-y²}-[(x²+y²)/2])  dy dx=\allowbreak 2Ö3p-[ 5/3]p

1.0.2  Double Integrals using polar coordinates

Direct Computations in polar coordinates  

1. Compute ò₀^p/2ò₁³  re^-r2 dr dq

   Inner: ò₁³  re^-r2 dr=\allowbreak -[ 1/2]e^-9+[ 1/2]e^-1  Using u=-r²  and du=-2r dr

   Completion: ò₀^p/2ò₁³  re^-r2 dr dq = \allowbreak -[ 1/4]e^-9p+[ 1/4]e^-1p

2. FInd the area bounded by the cardioid  r=1+sinq. 

   \diint_D1 dA=ò₀^2pò₀^{1+sin( q)} 1 r dr dq = \allowbreak [ 3/2]p

3. Find the area bounded by one leaf of the rose r=4cosq

   \diint_D1 dA=ò₀^pò₀^{4cos( q)}1 r dr dq = \allowbreak 4p

4. Find area inside both  r=1  and r=2sinq.

   \diint_D1 dA=2( ò₀^[(p)/6]ò₀^{2sin( q)} 1 r dr dq+ò_[(p)/6]^[(p)/2]ò₀¹1 r dr dq) = \allowbreak [ 2/3]p-[ 1/2]Ö3

Convert from Cartesian ( x,y) to polar coordinates before integrating  

1. Find \iint_Df( x,y)  dA  where D  is the region bounded by the x-axis, the line y=x and the circle x²+y²=1.

   \iint_Df( x,y)   dA=ò₀^[(p)/4]ò₀¹f( rcos( q) ,rsin( q) )  r dr dq

2. Find the volume of the solid bounded by the paraboloid z=4-x²-y²  and the xy-plane.

   V=ò_-2²ò_-Ö{4-x²}^Ö{4-x2}(4-x²-y²)  dy dx=ò₀^2pò₀²(4-r²)  r dr dq = \allowbreak 8p

3. Find the volume inside the sphere x²+y²+z²=25 and outside the cylinder x²+y²=9.

   V=\diint_D( Ö{25-x²-y²}-0) dA=2ò₀^2pò₃⁵Ö{25-r²} r dr dq = \allowbreak [ 256/3]p

4. Find the volume inside the sphere x²+y²+z²=25 and outside the cylinder ( x-1) ²+y²=1.

   [This is a project problem but a hint is to write the equation of the cylinder in polar coordinates.]