R, you'll get the volume of the sphere, which is 4/3 Pi*R^3. Integrating the surface area of the sphere wrt. Since the surface area of a sphere of radius R has units R^2, then the Surface area of a general sphere of radius R is 4*Pi*R^2. Integrate that between 0 and Pi and you'll get 4*Pi, which is the surface area of your sphere. Assuming your sphere has radius 1, you'll find the circumference of your circle r units away from a pole is 2*Pi*sin(r). Now how could you calculate the surface area of a sphere? If you get a basket ball or something you can see that the surface area of a sphere is the infinite sum of circles which starting from one pole of the surface of the sphere, get bigger, until one reaches the equator then shrink back to zero radius at the other pole. R you get 2*Pi*R, which is the circumference, and when you integrate the circumference you get the area. Can you see that the area of a circle is the infinite sum of smaller and smaller circles, until you reach the center of the circle? This is why when you differentiate the area of a circle Pi*R^2, wrt. Then you would sum an infinite number (ie.integrate) of 1d objects, in this case the circumference of the circle between 0 and R. Take for example, if you wanted to calculate the area of a 2d object such as a circle with radius R. To find the "volume" of a 4d object you got to integrate over a 3 dimensional surface.
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