In general, if a curve is always decreasing in an interval, using left endpoints for the riemann sum will give an overestimation of the area under the curve, whereas right endpoints will give an underestimation. On the other hand, if a curve is always increasing in an interval, using left endpoints for the riemann sum will give an underestimation of the area under the curve, whereas right endpoints will give an overestimation.

When you use a Riemann sum to approximate the area under the curve, you're just sketching rectangles under the curve, taking the area of each rectangle, and then adding the areas together. When you use midpoints, it means that you draw the rectangles such that the midpoints of their top edges touch the curve.

When you use a Riemann sum to approximate the area under the curve, you're just sketching rectangles under the curve, taking the area of each rectangle, and then adding the areas together. When you use right endpoints, it means that you draw the rectangles such that their upper-right corners touch the curve.

When you use a Riemann sum to approximate the area under the curve, you're just sketching rectangles under the curve, taking the area of each rectangle, and then adding the areas together. When you use left endpoints, it means that you draw the rectangles such that their upper-left corners touch the curve.

You can write an expanded sum in collapsed summation notation by identifying the pattern among the terms in the expanded sum.

You can expand a sum by plugging in each of the index numbers and taking the sum of the results.

With a non-infinite sum, finding its value is as simple as plugging in each of the index numbers and taking the sum of the results.

Marginal cost, revenue and profit represent the rate of change (the derivative) of the cost, revenue and profit, respectively.

Determine the speed and velocity of an object dropped from a roofwith this vertical motion problem.

We can find position, speed and velocity of an object thrown straight up from the ground.