To find the area under the curve of a function f(x) between an upper limit b, and a lower limit a,[br]we can use the definite integral.[br]This applet shows that using rectangular slices of small width, either completely under the curve, [br]or partially protruding from the curve, we can estimate the area under the curve by adding up the[br]area of the rectangular slices.[br][br]The definite integral is actually the exact solution of the area of the rectangular slices when the upper and lower estimate area of the rectangular slices become the same value. [br]This happens when the number of slices are increased to infinitely large numbers and the width of the rectangular slices are infinitestimally small.

1. Set the function and the upper and lower limits first.[br]2. Choose a small number of divisions or slices.[br]3. Click on Under Curve Slices checkbox to see the estimated area of all the slices (lower estimate)[br]4. Move the pointer Pt (blue colored diamond shape) to the slices and see the calculation of each rectangular slice.[br]5. Add the areas of each slice to confirm that the estimated area of all slices (lower estimate) is correctly shown[br]6. Repeat the same for Protruding Slices checkbox for Steps 3 to 5, for estimated area of all slices (upper estimate)[br]7. Move the pointer Pt (blue colored diamond shape) out of the graph area (So it will not show text for calculations.[br]8 . Click on the Show Definite Integral checkbox. Note the actual area under the curve between the limits.[br]9. Now use the numberofslices slider to increase the number of slices from 3 slowly to 20. [br] What do you observe about the 3 values of the areas (lower estimate, upper estimate and the definite integral?[br]10. Increase the number of slices from 20 to 100. What do you notice about the areas again? What can you conclude[br]11. Click on the Activity Conclusion checkbox to compare your conclusion with this