Approximating Areas with Riemann Sums

What is the primary purpose of using Riemann sums in numerical integration?

• To approximate the area under a curve

• To find the exact value of an integral

• To solve differential equations

• To calculate the derivative of a function

Which of the following is true about Riemann sums?

• They can approximate the area under a curve by summing the areas of rectangles.

• They can only be used with continuous functions.

• They provide an exact value for the area under a curve.

• They are more accurate than the Trapezoidal Rule and Simpson's Rule for all functions.

It is an approximate area of a region, obtained by adding up the areas of multiple simplified slices of the region.

• Riemann sum

• definite integral

• summation notation

• sum of a series

What does this picture represent?

• 
  Left Riemann Sum

• Middle Riemann Sum

• Right Riemann Sum

• Trapezoidal Sum

A Riemann Sum uses rectangles to

• approximate the area under a curve. The more rectangles, the better the approximation.

• approximate the area under a curve. The less rectangles, the better the approximation.

• approximate the area under a curve. The more rectangles, the worse the approximation.

Riemann Sum uses rectangles to

• approximate the area under a curve. The more rectangles, the better the approximation.

• approximate the area under a curve. The less rectangles, the better the approximation.

• approximate the area under a curve. The more rectangles, the worse the approximation.

Based on the table, use a Right Riemann sum and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up.)

• 5(3) + 1(4) + 2(5) + 1(7)

• 5(4) + 1(5) + 2(7) + 1(6)

• 5(3) + 6(4) + 8(5) + 9(7)

• 0(3) + 5(4) + 6(5) + 8(7)

• 35

• 40.625

• 32.1

• 41

Approximate the area between the x-axis and h(x) = 1/7-x from x = 2 to x = 5 using a left Riemann sum with 3 equal subdivisions.

• 37/60 units²

• 47/60 units²

• 75/81 units²

• 3/5 units²

Approximate the area between the x-axis and h(x) = 1/7-x from x = 2 to x = 5 using a left Riemann sum with 3 equal subdivisions.

• 37/60 units²

• 47/60 units²

• 75/81 units²

• 3/5 units²