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Additional exercises 5

Exercise

Evaluate the multiple integral I=Rf(x,y) dxdy by iterated integration, where:

  1. R=[0,1]×[0,1], f(x,y)=27xy(x+y)
  2. R=[0,1]×[0,1], f(x,y)=4x3
  3. R=[0,π/2]×[0,π/2], f(x,y)=7sin(x+y).
  4. R=[0,1]×[1,2], f(x,y)=xexy
  5. R=[0,1]×[1,2], f(x,y)=xexy

Exercise

Let R=[0,π]×[0,π].

Evaluate the following integral:

R(sinxsiny)2 dxdy

Hint: integrals of the form Rf(x)g(y) dxdy are equal to the product of integrals.

Exercise

Let SR2 be the triangular region with vertices (0,0), (π,0), (π,π). Evaluate integral

Sxcos(x+y) dxdy.

Exercise

Let SR2 be the region bounded by the curves y=sinx and the line segment {(x,0):x[0,π]}.

Sketch S and evaluate the integral

S(x2y2) dxdy.

Exercise

Let SR2 be the region bounded by the four curves x2y+8=0, x+3y+5=0, x=2 and x=4. Sketch S and find its centroid.

Exercise

Find the center of mass of trapezoidal object with corners (x,y)=(2,0), (2,4), (2,2) and (2,0) and density ρ(x,y)=x2+y+2.

Exercise

Use polar coordinates to evaluate

01[0xx2+y2 dy] dx.

Hint: sec3x dx=12(secxtanx+ln|secx+tanx|)+C.

Exercise

In this question we will evaluate the integral

I=V10x+8y+6z dxdydz

where the integral is over the half ellipsoid

V={(x,y,z):x24+y2+z21,x0}R3.

We choose a change of coordinates x=rcosθ, y=12rsinθ, z=z.

  1. Find the Jacobian of this transformation.
  2. Find the set of (r,θ,z) which corresponds to V.
  3. Using these, evaluate the integral I.

Exercise

The set V={(x,y,z):x2+y242,0z4x2+y2} is a cone of height 4 with base in the xy-plane.

The set W={(x,y,z):(x2)2+y24} is a cylinder.

Find the volume of the set DR3 which is the subset of the cone V which is contained within the cylinder W.

Hint: Use polar coordinates x=rcosθ, y=rsinθ to write the volume of D as

Tr(ar) drdθ.

for some aR, TR2.

Exercise

Let V be the solid bounded above by the sphere x2+y2+z2=4 and below by the cone z=x2+y2.

Find the volume of V.