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Slices | thickness | White Stack | volume | Blue Stack | volume | ΔV |
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Voice Over 1
Voice Over 2
This animation hopes to be useful in understanding the basics of integration. The idea is to show that integration is a special kind of summation.

Two stacks of four blocks with square faces and thickness 1 unit each on either side of the pyramid shown in the middle. The slab on the top of white stack has a side of 1 unit and the slabs below it have sides increasing in steps of 1 unit. That is, the topmost slab has a side of 1 unit and the one below it has a side of 2 units and so on.The slab on the bottom of the blue stack has a side of 5 units and the slabs above it have sides decreasing in steps of 1 unit. That is, the bottommost slab has a side of 5 units and the one above it has a side of 4 units and so on. The pyramid in the middle has a continuously varying side and has a top face of side 1 unit and bottom face of side 5 units. Our interest is to calculate the volume of this pyramid. We do this by calculating the sum of the volumes of the labs in the stack on the left and the stack on the right. Each slabs volume is simple to calculate, it is area times the thickness of the slab. The volume of the white stack will be lower than the volume of the pyramid and the voume of the blue stack will be higher than the volume of the pyramid. Suppose the four slabs are replaced by eight of them whoose sides are in steps of 0.5. The total thicknes of the stack remains the same. We calculate the volumes of the stacks on the right and left and we find the values getting closer. The volume of the white stack is the lower estimate and volume of the blue stack is the upper estimate. This can be repeated as we increase the number of slabs, keeping their total height same. As they get thinner the step wise stacks look almost like the pyramid in the middle and we end up getting a good approximation to the volume of the pyramid. It lies between two values of the volumes of the white stack and volume of the blue stack and these can be made very close by increasing the number of slices. Table below shows the same. Notice the decrease in the difference in volumes (shown in last column) of the blue and white stacks with increase in the number of sllices.

Two stacks of four blocks with square faces and thickness 1 unit each on either side of the pyramid shown in the middle. The slab on the top of white stack has a side of 1 unit and the slabs below it have sides increasing in steps of 1 unit. That is, the topmost slab has a side of 1 unit and the one below it has a side of 2 units and so on.The slab on the bottom of the blue stack has a side of 5 units and the slabs above it have sides decreasing in steps of 1 unit. That is, the bottommost slab has a side of 5 units and the one above it has a side of 4 units and so on. The pyramid in the middle has a continuously varying side and has a top face of side 1 unit and bottom face of side 5 units. Our interest is to calculate the volume of this pyramid. We do this by calculating the sum of the volumes of the labs in the stack on the left and the stack on the right. Each slabs volume is simple to calculate, it is area times the thickness of the slab. The volume of the white stack will be lower than the volume of the pyramid and the voume of the blue stack will be higher than the volume of the pyramid. Suppose the four slabs are replaced by eight of them whoose sides are in steps of 0.5. The total thicknes of the stack remains the same. We calculate the volumes of the stacks on the right and left and we find the values getting closer. The volume of the white stack is the lower estimate and volume of the blue stack is the upper estimate. This can be repeated as we increase the number of slabs, keeping their total height same. As they get thinner the step wise stacks look almost like the pyramid in the middle and we end up getting a good approximation to the volume of the pyramid. It lies between two values of the volumes of the white stack and volume of the blue stack and these can be made very close by increasing the number of slices. Table below shows the same. Notice the decrease in the difference in volumes (shown in last column) of the blue and white stacks with increase in the number of sllices.

No. of slices | thickness | lower estimate | upper estimate | difference |