Advanced MathFind the volume of the solid bounded by the elliptic cylinder 4x^2 + z^2 = 4 and the planes y = 0 and y = z + 2
11 months ago
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
1 Answers
Best Answer
Cmanuel1112 Staff answered 11 months ago
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
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