Da Nang-03/2014 Natural Science Department – Duy Tan University Lecturer: Ho Xuan Binh Triple Integrals in Cylindrical Coordinates In this section, we will learn about: Cylindrical coordinates and using them to solve triple integrals.

Triple Integrals in Cylindrical Coordinates Cylindrical Coordinates 1 Natural Science Department – Duy Tan University x = r cos θ y = r sin θ z = z r 2 = x 2 + y 2

Evaluating Triple InteGS. With CYL. CoorDS. 2 Natural Science Department – Duy Tan University Triple Integrals in Cylindrical Coordinates E = {(x, y, z) | (x, y) in D, u 1 (x, y) ≤ z ≤ u 2 (x, y)} D = {(r, θ) | α ≤ θ ≤ β, h 1 (θ) ≤ r ≤ h 2 (θ)}

Evaluating Triple Integrals 3 Natural Science Department – Duy Tan University Triple Integrals in Cylindrical Coordinates

Triple Integral In CYL. Coordinates. 4 Natural Science Department – Duy Tan University Writing x = r cos θ, y = r sin θ. Leaving z as it is. Replacing dV by r dz dr dθ. Triple Integrals in Cylindrical Coordinates

Example 1 5 Natural Science Department – Duy Tan University Evaluate where E is the region that lies inside the cylinder x 2 + y 2 = 16 and between the planes z = 3 and z = -5. Triple Integrals in Cylindrical Coordinates

Example 2 6 Natural Science Department – Duy Tan University Evaluate where E is enclosed by the planes z = 0 and z = x+y+5 and by the cylinders x 2 + y 2 = 4 and x 2 + y 2 = 9. Triple Integrals in Cylindrical Coordinates

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