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Approximating Areas
With Riemann Sums

Learn to estimate the area under a curve by partitioning intervals into rectangles: left, right, midpoint, and trapezoidal methods.

AP Calculus BC

What is the Area Under a Curve?

Given a continuous function \( f(x) \geq 0 \) on an interval \([a, b]\), the area under the curve is the region bounded by the graph of \( f \), the \( x \)-axis, and the vertical lines \( x = a \) and \( x = b \).

a b y x Area \(f(x)\)

Why Approximate the Integral?

Conceptual Foundation
Riemann sums define the definite integral. Understanding them deepens your grasp of what \(\int_a^b f(x)\,dx\) truly means.
Practical Need
Many real-world functions have no closed-form antiderivative (e.g., \(e^{-x^2}\)), so numerical approximation is essential.
AP Exam Skill
The AP Calculus BC exam frequently tests Riemann sum approximations using tables of values and graphical data.

Partitions and Subintervals

Divide \([a,b]\) into \(n\) equal subintervals. Each subinterval has width:

\(x_0 = a\) \(x_1\) \(x_2\) \(x_3\) \(x_4 = b\) \(\Delta x\)

In this example, \(n = 4\) subintervals. As \(n \to \infty\), the approximation converges to the exact integral.

The Four Riemann Sum Methods

Left Sum
Height of each rectangle is \(f(x_i)\), the left endpoint of the subinterval.
Right Sum
Height of each rectangle is \(f(x_{i+1})\), the right endpoint of the subinterval.
Midpoint Sum
Height of each rectangle is \(f\!\left(\frac{x_i + x_{i+1}}{2}\right)\), the midpoint.
Trapezoidal
Uses trapezoids instead of rectangles: average the left and right heights.

Left Riemann Sum

y x Left endpoints determine heights \(f(x)\)

Each rectangle's height is determined by the left endpoint of its subinterval. Notice the rectangles may overshoot or undershoot the curve.

Right Riemann Sum

y x Right endpoints determine heights \(f(x)\)

Identical structure to the left sum, but each rectangle's height comes from the right endpoint \(f(x_{i+1})\) instead.

Midpoint and Trapezoidal Rules

Midpoint Rule
$$M_n = \Delta x \sum_{i=0}^{n-1} f\!\left(\frac{x_i + x_{i+1}}{2}\right)$$
Uses the function value at the center of each subinterval. Often more accurate than left or right sums.
Trapezoidal Rule
$$T_n = \frac{\Delta x}{2}\Big[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)\Big]$$
Averages left and right sums: \(T_n = \frac{L_n + R_n}{2}\).

Over or Underestimate?

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

How to Compute a Riemann Sum

1
Identify the interval and \(n\). Find \(a\), \(b\), and the number of subintervals \(n\).
2
Compute \(\Delta x\). Calculate \(\Delta x = \frac{b - a}{n}\).
3
List the sample points. Determine each \(x_i\) (left, right, or midpoint depending on the method).
4
Evaluate \(f\) at each sample point. Compute \(f(x_i)\) for each subinterval.
5
Multiply and sum. Each area contribution is \(f(x_i)\cdot\Delta x\). Add them all to get the total approximation.

Example 1: Left Sum of \(f(x) = x^2\)

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

0 1 2 3 4 \(f(x) = x^2\)

Example 2: Right Sum of \(f(x) = x^2\)

Same integral: \(\displaystyle\int_0^4 x^2\,dx\) with \(n = 4\), but now using the right Riemann sum.

Notice how averaging the left and right sums dramatically improves accuracy. The trapezoidal rule is often a quick win on the AP exam.

Example 3: Table-Based Approximation

The table below gives selected values of a continuous function \(g\). Approximate \(\displaystyle\int_0^{12} g(x)\,dx\) using a left Riemann sum with the subintervals defined by the table.

\(x\) 0 3 7 10 12
\(g(x)\) 5 8 6 11 9

When subintervals have unequal widths, you must multiply each function value by its own \(\Delta x\). Do not assume a single \(\Delta x\) for all rectangles.

The Definite Integral as a Limit

This is the formal definition of the definite integral. Left, right, and midpoint sums are all special cases: they just choose different sample points \(x_i^*\).

Left: \(x_i^* = x_{i-1}\)
Uses the left boundary of the \(i\)-th subinterval.
Right: \(x_i^* = x_i\)
Uses the right boundary of the \(i\)-th subinterval.
Mid: \(x_i^* = \frac{x_{i-1} + x_i}{2}\)
Uses the midpoint of the \(i\)-th subinterval.

Common Mistakes to Avoid

✕ Wrong Number of Terms
Left sum uses \(n\) terms: \(f(x_0)\) through \(f(x_{n-1})\). Right sum also uses \(n\) terms: \(f(x_1)\) through \(f(x_n)\). Don't confuse indices.
✕ Assuming Equal Widths
When given a table, subintervals may have different widths. Always compute each \(\Delta x_i\) individually.
✕ Forgetting \(\Delta x\)
A Riemann sum is \(\sum f(x_i)\cdot\Delta x\), not just \(\sum f(x_i)\). The width factor is essential.

Summary of Methods

Method Formula Accuracy
Left Sum \(\Delta x\sum_{i=0}^{n-1} f(x_i)\) Least accurate
Right Sum \(\Delta x\sum_{i=1}^{n} f(x_i)\) Least accurate
Midpoint \(\Delta x\sum_{i=0}^{n-1} f(\bar{x}_i)\) More accurate
Trapezoidal \(\frac{\Delta x}{2}[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)]\) More accurate

Question 1 of 5

Using a left Riemann sum with \(n = 3\) equal subintervals, approximate \(\displaystyle\int_1^7 (2x+1)\,dx\).

Question 2 of 5

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

Question 3 of 5

Given the table of values for \(h(x)\):

\(x\) 0 2 5 8
\(h(x)\) 3 7 4 10

Approximate \(\displaystyle\int_0^8 h(x)\,dx\) using a right Riemann sum.

Question 4 of 5

The trapezoidal rule with \(n = 4\) subintervals on \([0,8]\) uses function values at \(x = 0, 2, 4, 6, 8\). The coefficients of \(f(0), f(2), f(4), f(6), f(8)\) in the formula are:

Question 5 of 5

A midpoint Riemann sum with \(n = 4\) on \([0,8]\) uses which sample points?