Before we look further at areas between curves and other applications, let's review the area under a curve and explore some extended-response questions.
Recall, that to find an area under a curve we can use:
- Approximation methods
- Geometric reasoning
- Definite integration
- Technology
Revisit our previous lesson on areas under a graph to review the rules and calculator instructions for approximation methods.
Consider a continuous function on the interval $\left[a,b\right]$[a,b], where $f\left(x\right)\ge0$f(x)≥0 for all $x$x in the interval. The area under the graph of $y=f\left(x\right)$y=f(x) from $x=a$x=a to $x=b$x=b can be approximated by subdividing the interval into $n$n rectangles of width $w=\frac{b-a}{n}$w=b−an and using one of the following methods:
Left endpoint approximation
| $A_L$AL | $=$= | $wf(a)+wf(a+w)+wf(a+2w)+wf(a+3w)\dots+wf(a+(n-1)w)$wf(a)+wf(a+w)+wf(a+2w)+wf(a+3w)…+wf(a+(n−1)w) |
| $=$= | $\sum_{k=1}^nwf(a+(k-1)w)$n∑k=1wf(a+(k−1)w) |
Right endpoint approximation
| $A_R$AR | $=$= | $wf(a+w)+wf(a+2w)+wf(a+3w)+wf(a+4w)\dots+wf(a+nw)$wf(a+w)+wf(a+2w)+wf(a+3w)+wf(a+4w)…+wf(a+nw) |
| $=$= | $\sum_{k=1}^nwf(a+kw)$n∑k=1wf(a+kw) |
For finding exact areas recall our lesson on definite integrals and revisit the calculator instructions found at the end of the lesson.
For a continuous function, $f(x)$f(x), on an interval $[a,b]$[a,b], the signed area enclosed by the graph $y=f(x)$y=f(x), the $x$x-axis and the lines $x=a$x=a and $x=b$x=b, is given by the definite integral:
| $\int_a^bf\left(x\right)dx$∫baf(x)dx | $=$= | $\left[F\left(x\right)\right]_{x=a}^{x=b}$[F(x)]x=bx=a |
| $=$= | $F\left(b\right)-F\left(a\right)$F(b)−F(a) |
If $f(x)\ge0$f(x)≥0 for all $x$x in $\left[a,b\right]$[a,b] then the signed area and the area of the region will be equal.
When finding the area bound by a function and the $x$x-axis over an interval where the graph has sections that fall below the $x$x-axis, we can choose to break the function into these separate sections, then calculate the area of positive and negative sections individually before summing the absolute value of each section.
Alternatively, we can use our calculator to integrate the absolute value of the function. That is, the area between the function $f\left(x\right)$f(x) and the $x$x-axis on the interval $a\le x\le b$a≤x≤b is:
Area $=$=$\int_b^a\left|f\left(x\right)\right|dx$∫ab|f(x)|dx
Tip: keep a list of common integrals handy, such as:
| Function $f\left(x\right)$f(x), $a\ne0$a≠0 | Integral $\int f\left(x\right)dx$∫f(x)dx |
|---|---|
| $ax^n$axn | $\frac{ax^{n+1}}{n+1}+C$axn+1n+1+C, $n\ne-1$n≠−1. |
| $e^{ax+b}$eax+b | $\frac{1}{a}e^{ax+b}+C$1aeax+b+C |
| $\cos\left(ax+b\right)$cos(ax+b) | $\frac{1}{a}\sin\left(ax+b\right)+C$1asin(ax+b)+C |
| $\sin\left(ax+b\right)$sin(ax+b) | $-\frac{1}{a}\cos\left(ax+b\right)+C$−1acos(ax+b)+C |
| $\left(ax+b\right)^n$(ax+b)n | $\frac{\left(ax+b\right)^{n+1}}{a\left(n+1\right)}+C$(ax+b)n+1a(n+1)+C, $n\ne-1$n≠−1. |
Use the following applet to explore the area bound by a curve and the $x$x-axis.
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Practice questions
Question 1
The graph below shows the function $f\left(x\right)=e^x$f(x)=ex.
Find an estimate for the area between the function and the $x$x-axis for $0\le x\le4$0≤x≤4 using the left-endpoint approximation and $4$4 rectangles.
Round your answer to three decimal places.
Find an estimate for the area between the function and the $x$x-axis for $0\le x\le4$0≤x≤4 using the right-endpoint approximation and $4$4 rectangles.
Round your answer to three decimal places.
Using technology, find an estimate for the area between the function and the $x$x-axis for $0\le x\le4$0≤x≤4 using the left-endpoint approximation and $50$50 rectangles.
Round your answer to four decimal places.
Use integration to find the exact area between the function and the $x$x-axis for $0\le x\le4$0≤x≤4.
Calculate the percentage error of the approximation in part (c) compared to the exact area.
Round your answer to the nearest percent.
Question 2
The graph of $y=\cos x$y=cosx is shown below. Find the exact value of $k$k such that ratio $A:B$A:B is $1:1$1:1.
Question 3
Consider the function $f\left(x\right)=6ax-6x^2$f(x)=6ax−6x2, for $a\ge0$a≥0.
For $a=1$a=1, sketch the function below.
Loading Graph...Find the area bound by the curve and the $x$x-axis.
Use your calculator to fill in the table below of the area bound by the graph and the $x$x-axis for different values of $a$a.
$a$a $1$1 $2$2 $3$3 $5$5 $10$10 Area (units2) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ What is the exact value of the area bounded by $f\left(x\right)=6ax-6x^2$f(x)=6ax−6x2 and the $x$x-axis, for $a\ge0$a≥0?