1.1: Areas Between Curves
- Page ID
- 163247
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A Review of Necessary Prerequisite Theory
I recommend watching the following video before our first class meeting as it is assumed you already know all of this theory from your Calculus I course (and we will be using it quite frequently).
An Aside: Measuring Distances
- Compute horizontal and vertical distances.
Area of a Region between Two Curves
Let's derive the following theorem.
Let \( f(x)\) and \( g(x)\) be continuous functions such that \( f(x) \geq g(x)\) over an interval [\( a,b]\). Let \(\textbf{R}\) denote the region bounded above by the graph of \( f(x)\), below by the graph of \( g(x)\), and on the left and right by the lines \( x=a\) and \( x=b\), respectively. Then, the area of \(\textbf{R}\) is given by\[A=\int ^b_a[f(x)−g(x)]dx. \nonumber \]
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle. Find the area of the region.\[ x=7y-y^2, \quad x= \dfrac{1}{2} y - \sin(y), \quad y=3, \quad y=6 \nonumber \]
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle. Find the area of the region.\[ x-2y^2 \geq 0, \quad 1-x-|y| \geq 0 \nonumber \]
Being asked to evaluate an integral and being asked to find the area between two curves are two completely different questions.
The former is without context and can often be computed by finding antiderivatives.
On the other hand, the latter involves making decisions about when one curve is "above" the other and building integrals accordingly. We will see this in the following example.
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle. Find the area of the region.\[ y = \tan\left( 3x \right), \quad y = 2\sin\left( 3x \right), \quad -\dfrac{\pi}{9} \leq x \leq \dfrac{\pi}{9} \nonumber \]
Areas of Compound Regions
Let's build a justification for the following theorem.
Let \( f(x)\) and \( g(x)\) be continuous functions over an interval \( [a,b]\). Let \(\textbf{R}\) denote the region between the graphs of \( f(x)\) and \( g(x)\), and be bounded on the left and right by the lines \( x=a\) and \( x=b\), respectively. Then, the area of \(\textbf{R}\) is given by\[A=\int ^b_a|f(x)−g(x)|dx. \nonumber \]
Compute and interpret in terms of areas.\[ \int_{-2}^2 \left|8^x - 5^x \right| \, dx \nonumber \]
Synthesis Questions
Find the area of the region bounded by the parabola \(y=x^2\), the tangent line to this parabola at \(x=1\), and the \(x\)-axis.
Find the number \(b\) such that the line \(y=b\) divides the region bounded by the curves \(y=x^2\) and \(y=4\) into two regions with equal area.