Mathematica:  Project 3

1)	Evaluate the given iterated integrals and plot the solid corresponding to each one.
		a)	 (∫_0^1)(∫_0^(x^2))(4-(x^2)+(y^2))dydx		b)	 (∫_0^4)(∫_0^(Sqrt[2-(y^2)]))(x^2) ydxdy		c)	 (∫_0^1)(∫_0^x)(y/(1+x y))dydx

Integrate[Integrate[4-x^2+y^2,{y,0,x^2}],{x,0,1}]
Integrate[Integrate[x^2y,{x,0,Sqrt[2-y^2]}],{y,0,4}]
Integrate[Integrate[y/(1+xy),{y,0,x}],{x,0,1}]

2)	 Evaluate the given double integrals and plot the solid corresponding to each one.
		a)	∫(∫_D)(x+y)dA,  D={(x,y):0≤x≤3,0≤y≤(Sqrt[x])} 	
		
		b)	∫(∫_D)(Sqrt[x+y])dA, D={(x,y):0≤x≤1-(y^2),0≤y≤1}
		
		c)	∫(∫_D)(e^(x+y))dA, where D={(x,y):(x^2)+(y^2)≤4}

Integrate[Integrate[x+y,{y,0,Sqrt[x]}],{x,0,3}]
Plot3D[x+y,{x,0,3},{y,0,Sqrt[3]}]

Integrate[Integrate[Sqrt[x+y],{x,0,1-y^2}],{y,0,1}]
Plot3D[Sqrt[x+y],{x,0,1},{y,0,1}]

Integrate[Integrate[E^(x+y),{x, -Sqrt[4-y^2], Sqrt[4-y^2]}],{y, -2, 2}]
Plot3D[E^(x+y),{x, -2, 2},{y, -2, 2}]

3) 	Find the volume of the given solid S bounded under the paraboloid z=16-(x^2)-(y^2) and above the region bounded between the line y=x and the parabola y=6-(x^2).

Integrate[Integrate[16-x^2-y^2,{y,x,6-x^2}],{x,0,6}]

4)	Evaluate the given iterated integrals:
	a)	(∫_0^1)(∫_0^x)(∫_0^(y^2))(x+y+z)dzdydx		b)	 (∫_(-1)^1)(∫_(-(Sqrt[1-(x^2)]))^(Sqrt[1-(x^2)]))(1-(x^2)-(y^2))dydx

Integrate[Integrate[Integrate[x+y+z, {z,0,y^2}],{y,0,x}],{x,0,1}]
Integrate[Integrate[1-x^2-y^2,{y,-Sqrt[1-x^2],Sqrt[1-x^2]}],{y,-1,1}]

5)	 Evaluate the given triple integrals:
		a)	 ∫∫(∫_W)(x+y z)dV, where W={(x,y,z):0≤x≤1,0≤y≤(Sqrt[x]),0≤z≤(y^2)}.
		
		b)	 ∫∫(∫_W)sin ydV, where W lies under the plane z=1+x+y and above the triangular region bounded by x=0, x=2, and y=3x.

Integrate[Integrate[Integrate[x+yz,{z,0,y^2}],{y,0,Sqrt[x]}],{x,0,1}]
Integrate[Integrate[Integrate[Sin[y],{z,0,1+x+y}],{y,0,3x}],{x,0,2}]

6)	  Find  (∫_C)f(x,y,z)ds, where  f(x,y,z)= x (y^(2 ))-4 zy and C is given by  x= 2t, y= (t^(2/3)), and z=1-3(t^2), for 0≤t≤1.

7)	 Find  (∫_C)F(x,y)·ds, where    F(x,y)=⟨ (e^(3x-2y)),(e^(2x+3y))⟩ and C is given by  x= 2t, y= sin t,  0≤t≤ π

8)	 Find  (∫_C)F(x,y,z)·ds, where  F(x,y,z)=⟨xyz,-xz,xy ⟩ and C is given by  x= t, y= 2(t^2), z=3t  0≤t≤ 1

9)	 Find ∫(∫_S)F·dS, where F(x,y,z)=⟨ (e^z),z,y x⟩ and S is given by G(u,v)=(u v,u-v,u),  0≤u≤2, and -1≤v≤1, and oriented by n=(Subscript[T, u])×(Subscript[T, v]).

10)	 Find the curl of the  vector field  F(x,y,z)=⟨ ln ((x^2)+(y^2)+(z^2)),x/ z ,(e^x)sin (y z)⟩

11)  Use Stokes's Theorem to compute the flux of the curl of the vector field F(x,y,z)=⟨ tan(x y z),(e^(y-x z)),sec((y^2)x)⟩ cross the surface S, where S the upper hemisphere of radius 4.

12)	 Find the divergence of the given vector field  F(x,y,z)=⟨ x y z,(x^2)+(y^2)+(z^2),x y+y z+x z⟩

13)	 Use the Divergence Theorem to calculate the flux of the vector field F across the surface S, where  F(x,y,z)=⟨ x (e^(z )),(y^2),y+z x⟩ and S is tetrahedron bounded by the plane 3x+4y+5z=15 and the coordinate planes in the first octant.