   Chapter 15.7, Problem 18E

Chapter
Section
Textbook Problem

Use cylindrical coordinates.18. Evaluate ∭ E Z   d V , where E is enclosed by the paraboloid z = x2 + y2 and the plane z = 4.

To determine

To evaluate: The given triple integral by using cylindrical coordinates.

Explanation

Given:

The function is f(x,y,z)=z .

The region E is enclosed by the paraboloid z=x2+y2 and the plane z=4 .

Formula used:

If f is a cylindrical region E given by h1(θ)rh2(θ),αθβ, u1(rcosθ,rsinθ)zu1(rcosθ,rsinθ) where 0βα2π , then,

Ef(x,y,z)dV=αβh1(θ)h2(θ)u1(rcosθ,rsinθ)u2(rcosθ,rsinθ)f(rcosθ,rsinθ,z)rdzdrdθ (1)

If g(x) is the function of x and h(y) is the function of y and k(z) is the function of z, then,

abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (2)

The cylindrical coordinates (r,θ,z) corresponds to the rectangular coordinates (x,y,z) is,

r=x2+y2θ=tan1(yx)z=z

Calculation:

Solve the given equations,

z=x2+y2z=r2r2=4r=2

It is observed that r varies from 0 to 2, θ varies from 0 to 2π and z varies from r2 to 4. Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the value of the given triple integral is,

Ef(x,y,z)dV=0202πr24z(r)dzdθdr=0202πr24zrdzdθdr

Use the equation (2) to integrate and apply the limit values

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