Areas and Distances

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The Area Problem and Examples

Integration is a process for computing bulk quantities by estimating and adding up smaller pieces. The mantra (and the 5th Pillar of Calculus) is:

the whole is the sum of the parts

The idea works in many different settings. In this learning module we will develop the simplest application of integration, namely finding the area under a curve.

Example 1: Find the area under the curve $y=2$ and above the $x$-axis between $x=1$ and $x=4$.

Solution: The region is just a rectangle. The height of the rectangle is $2$, and the width is $4-1=3$, so the area is 6.

Example 2: Find the approximate area under the curve $y=f(x)$ and above the $x$ axis between $x=1$ and $x=4$, where $f(x) = 2 + 0.02 x$.

Solution: This is almost the same problem, except that the height of the curve isn't constant. Even so, the height doesn't change much between $x=1$ and $x=4$, so we can replace the curve $y=f(x)$ with the horizontal line $y=f(1)$ to get an approximate total area of $(4-1)f(1) = 3(2.02)=6.06$.

The answer in Example 2 is a little bit too small, since the curve goes up from $y=2.02$ at $x=1$ to $y=2.08$ at $x=4$. If we wanted an overestimate of the area, we would use the value of $f(x)$ at the right endpoint instead: $(4-1)f(4)=3(2.08)=6.24$. And if we wanted a more accurate guess, we might use the midpoint $f(2.5)=2.05$ and estimate the area as $(4-1)f(2.5)=3(2.05)=6.015$. But whether you use the left endpoint, the right endpoint, or the midpoint, you get roughly the same answer, since the function just isn't changing very much between $x=1$ and $x=4$.

So how do you find the area under a curve that is changing a lot? For instance, how would you find the area under $y=x^2$ between $x=-1$ and $x=1$? The answer is to break the interval $[-1,1]$ up into pieces, each so small that the function doesn't change much in that piece. Then add up the pieces to get a good estimate of the total. As with most of calculus, the exact answer is then the limit of these approximate answers.

The video on the right shows how this area can be approximated by adding up narrower and narrower rectangles, with $x_i^*$ being the left endpoint. (You may need to refresh this page and hit the 'replay' button a few times, since it's only 6 seconds long.)

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

$f$ vs. $f'$

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

The Area Problem and Examples

Main Ideas and Examples