A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R (> r) from the center of the circle. Find the volume of the torus.

Consider a circle with radius centered at some point on the -axis. This circle has equation

Revolve the region bounded by this circle across the -axis to get a torus. Using the shell method, the volume of the resulting torus is

where .

So the volume is

Substitute

and the integral becomes

Notice that is an odd function, so the integral over is 0. This leaves us with

Write

so the volume is

pentagon

step-by-step explanation:

the coordinates   a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.

probably the last option

step-by-step explanation: