01 / 21

Approximating Areas
With Riemann Sums

Learn how to estimate the area under a curve by partitioning an interval into rectangles and summing their areas.

AP Calculus AB

Why Approximate Area Under a Curve?

In calculus, we often need to find the exact area between a function and the x-axis. But most functions don't have neat geometric formulas for their enclosed regions. Riemann sums give us a powerful way to approximate that area using simple shapes: rectangles.

What Is a Riemann Sum?

A Riemann sum divides the interval \([a, b]\) into \(n\) subintervals, builds a rectangle on each subinterval, and adds up all the rectangle areas. As \(n\) grows, the approximation gets closer to the true area.

x y a b
\(\Delta x\)
\(n = 4\) subintervals

Key Vocabulary

Partition
Dividing \([a,b]\) into \(n\) equal subintervals of width \(\Delta x\).
Sample Point
The x-value in each subinterval used to determine rectangle height.
Subinterval Width
\(\Delta x = \dfrac{b - a}{n}\), the uniform width of each rectangle.

The Riemann Sum Formula

The choice of sample point \(x_i^*\) determines the type of Riemann sum: left endpoint, right endpoint, or midpoint.

Three Types of Riemann Sums

Left Sum Underestimate (increasing f) Right Sum Overestimate (increasing f) Midpoint Sum Usually most accurate

Overestimate vs. Underestimate

Whether a Riemann sum overestimates or underestimates the true area depends on two factors: the type of sum (left or right) and whether \(f\) is increasing or decreasing.

Left Sum Right Sum
\(f\) is increasing Underestimate Overestimate
\(f\) is decreasing Overestimate Underestimate

How to Set Up a Riemann Sum

1
Identify \(a\), \(b\), and \(n\). These give you the interval and the number of rectangles.
2
Compute \(\Delta x = \frac{b - a}{n}\). This is the width of each rectangle.
3
List the sample points \(x_i^*\). Use left endpoints \(x_i = a + i\,\Delta x\) for \(i = 0, 1, \ldots, n-1\), right endpoints \(x_i = a + i\,\Delta x\) for \(i = 1, \ldots, n\), or midpoints.
4
Evaluate \(f(x_i^*)\) at each sample point to get rectangle heights.
5
Sum up: \(S_n = \sum f(x_i^*)\,\Delta x\). Multiply each height by \(\Delta x\) and add.

Example: Left Riemann Sum

Approximate \(\displaystyle\int_0^4 x^2\,dx\) using a left Riemann sum with \(n = 4\).

x y 0 1 2 3 4 0 1 4 9

Example: Right Riemann Sum

Now approximate \(\displaystyle\int_0^4 x^2\,dx\) using a right Riemann sum with \(n = 4\).

0 1 2 3 4 1 4 9 16

Example: Midpoint Sum

Approximate \(\displaystyle\int_0^4 x^2\,dx\) using a midpoint Riemann sum with \(n = 4\).

Riemann Sums from a Table

On the AP exam, you may be given a table of values instead of a formula. You apply the same process: multiply each function value by \(\Delta x\) and sum.

\(x\) 0 2 4 6 8
\(f(x)\) 3 7 11 9 5

Follow the salt through a working mixing rig

A stirred transparent tank has one upper inlet and one lower outlet with a valve. The exact diagram links its 100 L salt balance to a computed approach to equilibrium.Keep the unequal interval widths in a right-endpoint sum: 2×7 + 1×9 + 2×6 + 3×3 estimates 44 L of accumulated flow.

Nonuniform right endpoint sum

Keep the unequal interval widths in a right-endpoint sum: 2×7 + 1×9 + 2×6 + 3×3 estimates 44 L of accumulated flow.Keep the unequal interval widths in a right-endpoint sum: 2×7 + 1×9 + 2×6 + 3×3 estimates 44 L of accumulated flow.

More Rectangles = Better Accuracy

\(n = 2\)
\(n = 4\)
\(n = 8\)
 Increasing accuracy

Sigma Notation for Riemann Sums

On the AP exam, you must be comfortable writing and interpreting Riemann sums in summation notation.

Left Sum
$$L_n = \sum_{i=0}^{n-1} f(a + i\,\Delta x)\,\Delta x$$
Right Sum
$$R_n = \sum_{i=1}^{n} f(a + i\,\Delta x)\,\Delta x$$
Midpoint Sum
$$M_n = \sum_{i=1}^{n} f\!\left(a + \left(i - \tfrac{1}{2}\right)\Delta x\right)\Delta x$$

Common Mistakes

Wrong Endpoints
Using right endpoints when asked for left, or vice versa. Always check which type is requested.
Forgetting \(\Delta x\)
Summing function values without multiplying by the width: area = height \(\times\) width.
Unequal \(\Delta x\)
When a table has unequal spacing, you can't use a single \(\Delta x\). Multiply each \(f(x_i^*)\) by its own width.

Summary Comparison

Property Left Right Midpoint
Sample point \(a + i\,\Delta x\) \(a + i\,\Delta x\) \(a + (i-\tfrac{1}{2})\Delta x\)
Index range \(i = 0, \ldots, n-1\) \(i = 1, \ldots, n\) \(i = 1, \ldots, n\)
Accuracy Lower Lower Higher
Increasing \(f\) Under Over Depends
Decreasing \(f\) Over Under Depends

Question 1 of 5

Using a left Riemann sum with \(n = 3\) subintervals, approximate \(\displaystyle\int_1^4 2x\,dx\).

Question 2 of 5

If \(f\) is a continuous, increasing function on \([a, b]\), which statement is true?

Question 3 of 5

A table gives values of \(g(x)\):

\(x\) 1 3 5 7
\(g(x)\) 4 10 8 2

What is the right Riemann sum approximation for \(\displaystyle\int_1^7 g(x)\,dx\) with 3 equal subintervals?

Question 4 of 5

The midpoint Riemann sum for \(\displaystyle\int_0^6 f(x)\,dx\) with \(n = 3\) uses which sample points?

Question 5 of 5

Which expression represents a right Riemann sum for \(\displaystyle\int_2^5 x^3\,dx\) with \(n\) subintervals?