Calculus II Workbook For Dummies

Chapter 4

Approximating Area with Riemann Sums

IN THIS CHAPTER

Approximating definite integrals with Riemann sums that use left and right rectangles

Using midpoint integrals to improve Riemann sums

Estimating area more precisely by replacing rectangles with trapezoids

Applying Simpson’s rule for approximating definite integrals

In Chapter 1, you used the definite integral to provide a precise way to state area problems on the xy-graph. You found that, in a few simple cases, you can use geometry to evaluate definite integrals. In most cases, however, solving an area problem requires more complex math than geometry offers.

In this chapter, you use Riemann sums to approximate the area under a function. To begin, you use left and right rectangles to give you a basic estimate of area. Next, you use midpoint rectangles to improve these estimates.

After that, you see how changing rectangles to trapezoids can further improve your area approximations. To finish, you write Riemann sums using Simpson’s rule to provide an even better approximation of the definite integral.

Calculating Riemann ...