Find the volume of the solid bounded by the elliptic cylinder 4x^2 + z^2 = 4 and the planes y = 0 and y = z + 2

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Best Answer

According to the given information, it is required to find the volume of the solid bounded by elliptic cylinder and the planes.
4x^2 + z^2 = 4 and the planes y = 0 and y = z + 2
Now find the limit and evaluate the integral.
y limit is form 0 to z + 2
z limit: 4x^2 + z^2 = 4 -> z = +- sqrt(4 - 4x^2)
x limit: 4:2 = 4 -> x = +- 1
volume = int_(-1)^(1) int_(-sqrt(4-4x^2))^(sqrt(4-4x^2)) int_(z+2)^(0) dydzdx
= int_(-1)^(1) int_(-sqrt(4-4x^2))^(sqrt(4-4x^2)) [y]_(0)^(z+2) dzdx
= int_(-1)^(1) int_(-sqrt(4-4x^2))^(sqrt(4-4x^2)) (z+2) dzdx
= int_(-1)^(1) [(z^2)/(2) + 2z]_(-sqrt(4-4x^2))^(sqrt(4-4x^2)) dx
= int_(-1)^(1) 2 - 2x^2 + 2 sqrt(4-4x^2) - (2 - 2x^2 - 2 sqrt(4-4x^2))dx
= int_(-1)^(1) 4sqrt(4-4x^2) dx
Solving further:
int_(-1)^(1) 4sqrt(4-4x^2) dx = 8 int_(-1)^(1) sqrt(1-x^2) dx
= 8 [(x)/(2) sqrt(1-x^2)+ (1)/(2) sin^(-1)(x)]_(-1)^(1)
= 8 [(1)/(2) sqrt0 + (1)/(2)sin^(-1) (1)[- (1)/(2) sqrt0 + (1)/(2) sin^(-1)(-1)]]
= 8 [(1)/(2)((pi)/(2))-[(1)/(2)(-(pi)/(2))]]
= 8 [(pi)/(4) + (pi)/(4)]
= 8 ((2pi)/(4))
Volume (V) = 4pi