Calculus Integration

Definite integrals (Riemann sums)

Q: What is a definite integral?

The exact signed area between a curve and the x axis from a to b, written the integral from a to b of f of x dx. It is the limit of Riemann sums.

Q: What is a Riemann sum?

An approximation of area by adding up rectangle areas, the sum of f of x i times delta x. As the rectangles get thinner, the sum approaches the definite integral.

Q: What does signed area mean?

Regions above the x axis count as positive area and regions below count as negative, so the integral is their difference.

Q: What is the difference between left and right Riemann sums?

They measure each rectangle's height at the left edge or the right edge of the strip. Both approach the same integral as the strips get thinner.

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Definite Integrals and Riemann Sums

California Calculus • Standard 13.0 • Integration

Definite Integrals and Riemann Sums is a topic in Integration in the California Calculus Standards. It is aligned to Standard 13.0, which requires students to know the definition of the definite integral by using Riemann sums.

The definite integral \(\int_a^b f(x)\,dx\) is the signed area between a curve and the \(x\)-axis. It is defined as the limit of Riemann sums — the total area of thin rectangles as they become infinitely thin.

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Theory

The definite integral \(\displaystyle\int_a^b f(x)\,dx\) is the exact signed area between a curve and the \(x\)-axis. It is defined as the limit of Riemann sums — the total area of thin rectangles as they become infinitely thin.

To measure the area under a curve, cut \([a,b]\) into \(n\) strips of width \(\Delta x\) and add up the rectangle areas. This total is a Riemann sum:

\[\sum_{i=1}^{n} f(x_i)\,\Delta x.\]

As \(n\to\infty\) (rectangles infinitely thin) the sum approaches the exact area, which is the definite integral:

\[\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x.\]

Key idea: the integral is a signed area — regions below the \(x\)-axis subtract. Left, right, and midpoint sums differ only in where each rectangle's height is measured.

A Riemann sum: rectangles approximating the area.

The exact area is \(\displaystyle\int_a^b f(x)\,dx\).

The definite integral as a limit of Riemann sums:

\[\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x,\qquad \Delta x=\dfrac{b-a}{n}\]

definite integral is the limit of the sum of f of x i times delta x

Area shortcuts: for a triangle or rectangle region you can read the integral off the geometry instead of summing.

How to estimate with a Riemann sum

Find \(\Delta x=\dfrac{b-a}{n}\) and the sample points.
Evaluate the height \(f(x_i)\) at each left, right, or midpoint.
Multiply and add: \(\sum f(x_i)\,\Delta x\). More rectangles give a better estimate.

Example 1 — A left Riemann sum

Estimate \(\displaystyle\int_0^2 x^2\,dx\) with two left rectangles.

Solution

Width \(\Delta x=1\); left heights at \(x=0,1\).

\(L_2\)	\(=\)	\(1\cdot f(0)+1\cdot f(1)\)
	\(=\)	\(0+1=1\)

The estimate is \(1\) (the exact value is \(\dfrac{8}{3}\)).

left Riemann sum estimate equals 1

Example 2 — Area as an integral

Evaluate \(\displaystyle\int_0^2 x\,dx\) as an area.

Solution

The region under \(y=x\) on \([0,2]\) is a triangle.

\(\int_0^2 x\,dx\)	\(=\)	\(\dfrac{1}{2}\cdot 2\cdot 2\)
	\(=\)	\(2\)

integral equals 2, the area of the triangle

Example 3 — A constant integrand

Evaluate \(\displaystyle\int_1^4 3\,dx\) as an area.

Solution

The region is a rectangle of width \(3\) and height \(3\).

\(\int_1^4 3\,dx\)

\(=\)

\(3\cdot 3=9\)

integral equals 9, area of the rectangle

Example 4 — Signed area

What does \(\displaystyle\int_a^b f(x)\,dx\) give when \(f<0\)?

Solution

Below the axis, each rectangle has a negative height.

The definite integral is a signed area: regions below the \(x\)-axis count as negative.

definite integral is signed area, negative below the axis

Common pitfalls

The integral is signed area. Area below the \(x\)-axis counts as negative, so it can be \(0\) even when the region is not empty.

Left and right sums differ. They use the height at the left or right edge of each strip; do not mix them up.

Include \(\Delta x\). A Riemann sum multiplies each height by the strip width, not just the heights.

Frequently asked questions

What is a definite integral?

The exact signed area between a curve and the \(x\)-axis from \(a\) to \(b\), written \(\int_a^b f(x)\,dx\). It is the limit of Riemann sums.

What is a Riemann sum?

An approximation of area by adding up rectangle areas \(\sum f(x_i)\,\Delta x\). As the rectangles get thinner, the sum approaches the definite integral.

What does signed area mean?

Regions above the \(x\)-axis count as positive area and regions below count as negative, so the integral is their difference.

What is the difference between left and right Riemann sums?

They measure each rectangle's height at the left edge or the right edge of the strip. Both approach the same integral as the strips get thinner.

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