Find the volume of the solid that lies within both the cylinder x^2+y^2=1 and the sphere x^2+y^2

Find the volume of the solid that lies within both the...

Key Concepts

Intersection of Surfaces

Determining the intersection of the cylinder and sphere requires analyzing where the equations of both surfaces are simultaneously satisfied. This intersection identifies the exact region over which the volume must be computed, and is essential for correctly setting the limits of integration in the chosen coordinate system.

Triple Integration

Triple integration is the process used to calculate the volume of a three-dimensional region. It involves setting up and evaluating an integral with appropriate bounds that precisely describe the intersection of the given surfaces, ensuring that only the points within both the cylinder and the sphere are included.

Cylinder

A cylinder in analytic geometry is a surface defined by an equation in two dimensions, typically representing a set of points equidistant from a given line (the axis of the cylinder). In problems like this, the cylinder’s equation (x² + y² = constant) describes a right circular cylinder with symmetry about the z-axis, providing a natural boundary for integration.

Sphere

A sphere is a three-dimensional surface where all points are equidistant from a fixed center point. The equation x² + y² + z² = constant describes a sphere, and in this context, it provides a curved boundary that intersects with the cylindrical region, defining the limits of the volume being calculated.

Cylindrical Coordinates

Cylindrical coordinates (r, ?, z) are often used in problems involving cylinders and spheres because they align with the symmetry of these shapes. This coordinate system simplifies the integration process, as the cylinder’s equation becomes r = constant and the sphere’s equation can be easily expressed in terms of r and z.

Transcript

00:01 Okay, so this problem wants you to find the volume of a region that is within a cylinder of radius 1 and a sphere of radius 2.

00:11 So over here on the right, i have a plot of the cylinder and the sphere.

00:15 And the region that we're going to be integrating is going to be this part.

00:19 That is within both the cylinder and the sphere, that region.

00:26 So to start off, the first thing we want to do is to calculate, is to convert everything into cylinder.

00:33 So let's do that all we have to do is convert dv into cylindrical coordinates which is just r times d z dr d theta now looking at our boundaries we have x plus y squared equals to 1 and we know that x plus y squared equals to r squared so we can rewrite this as r equals to 1 and for the sphere we have x squared plus y squared plus z squared equals to 4 and we can rewrite this as r squared plus z squared equals to 4 and we can subtract r squared from both sides and take the square root to isolate z to get z equals to 4 minus r squared.

01:16 All right.

01:17 So now that we have everything in cylindrical coordinates, let's figure out the boundaries.

01:21 So first up we have dz.

01:23 We know that dz, you know, it's from the bottom of the sphere to the top of the sphere, right? so it's going to be from negative z, negative square root of 4 minus r square to positive square root of 4 minus r square to positive square root of 4.

01:35 4 minus r square.

01:38 But, however, we can use something called symmetry.

01:42 As we know that the top half of the sphere and the bottom half of the sphere, or the regions, as i should say, those regions are identical.

01:53 So all we have to essentially do is integrate half of it and multiply it by 2.

01:57 So all we have to do is integrate from 0 to 4 minus r square, which is half of the region, and multiply it by 2.

02:06 All right, now we have to do dr, and we know that dr is the cylinder, the outside cylinder is the maximum radius of 1, and we know that it's a full rotation of 0 to 2 pi radiance.

02:23 So nothing changes there.

02:25 So now we have everything in cylindrical coordinates and we have boundaries so we can go ahead and calculate the integral.

02:33 So to start off, we have to integrate with respect to z.

02:38 And if we were to do that, we would end up with a double integral of, we end up a double integral of r times z from 0 to 4 minus r squared, drd theta.

02:58 All right.

02:59 And if we were to plug in those boundaries into the integral, we would get double integral of r times the square root of 4 minus r squared, dr, d theta.

03:18 All right, so now looking at this integral, we have to execute a u substitution...

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