Question

Suppose f(x,y,z) = x2 + y2 + z2 and W is the solid cylinder with height 7 and base radius 2 that is centered about the z-axis with its base at z = −2. Enter θ as theta.
A) As an iterated integral, ∭WfdV = ∫BA∫DC∫FE dzdrdθ with limits of integration.
B) Evaluate the integral.

Answer 1

Answer:

A)

A = 0       B = 2π

C = 0       D = 2

E = - 2      F = 5

B)

V = 216 π

Step-by-step explanation:

Using cylindrical coordinates

x= rcosθ , y = rsinθ, z = z

Then, dV =rdzdrdθ

where x² + y² + z² = r² + z²

E= ( 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 2, -2 ≤ z ≤ 5 )

∴

A = 0       B = 2π

C = 0       D = 2

E = - 2      F = 5

$V = \int \int \limits_ E \int \ dV \\ \\ \\ V = \int \limits ^{2 \pi}_{0} \int \limits ^{2 }_{0} \int \limits ^{5}_{2} (r^2 + z^2) \ dzdrd \theta \\ \\ \\ V = \int \limits ^{2 \pi}_{0} \int \limits ^{2 }_{0} \begin {bmatrix} zr^3 + \dfrac{rz^3}{3} \end {bmatrix}^5_{-2} \ dr d\theta$

$V = \int \limits ^{2 \pi}_{0} \int \limits ^{2 }_{0} \begin {bmatrix} 5r^3 + \dfrac{125 \ r}{3} + r^3 + \dfrac{r}{3} \end {bmatrix} \ dr d\theta$

$V = \int \limits ^{2 \pi}_{0} \int \limits ^{2 }_{0} \begin {bmatrix} 6r^3 + \dfrac{126 \ r}{3} \end {bmatrix} \ dr d\theta$

$V = 6 \int \limits ^{2 \pi}_{0} \int \limits ^{2 }_{0} \begin {bmatrix} r^3 + \dfrac{21 \ r}{3} \ \end {bmatrix} \ dr d\theta$

$V = 6 \int \limits ^{2 \pi}_{0} \begin {bmatrix} \dfrac{r^4}{4} + \dfrac{21 }{6} \ r^2 \end {bmatrix}^2_0 \ d\theta$

$V = (2 \pi) 6 \begin {bmatrix} \dfrac{16}{4} + \dfrac{21 }{6} (4) \end {bmatrix}$

V = 12π (4 + 14)

V = 12π(18)

V = 216 π