Final answer:
To calculate the volume of the solid bounded by the cylinder x²+z²=4 and the planes y=-1 and y+z=4, one must set up a triple integral in cylindrical coordinates. After defining the limits for r, theta, and y based on the given equations, the volume can be found by evaluating the triple integral with these limits.
Step-by-step explanation:
Using Triple Integrals to Find Volume
To find the volume of the solid enclosed by the cylinder x²+z²=4 and the planes y=-1 and y+z=4, one can use a triple integral. In this case, the integral will be set up in cylindrical coordinates because the equation of a cylinder is given, which suggests symmetry about one of the principal axes. First, rewrite the cylinder's equation in terms of cylindrical coordinates: r² = 4 where r is the radial distance in the xz plane and the limits for r will be from 0 to 2.
Next, determine the limits for θ (theta), which are straightforward since the cylinder is symmetrical about the axis, hence θ will range from 0 to 2π. Then, analyze the plane equations to find the limits for y. The plane y=-1 provides the lower bound for y, whereas the plane y+z=4 can be rearranged to y=4-z or y=4-rsin(θ) in cylindrical coordinates, which is the upper bound for y.
The volume integral in cylindrical coordinates thus becomes:
∫∫∫_V r dy dθ dr
where the limits of integration for r are [0, 2], for θ are [0, 2π], and for y are [-1, 4-rsin(θ)]. By evaluating this triple integral, one can obtain the volume of the solid.