15D.3 - Area between two curves
Similar to mixed regions, the curves where the area is bound between can cross meaning that one curve is not the "upper" curve for the entire interval \(x\in [a,b]\). When this occurs, separate integrals can be constructed between the points of intersection with the correct upper and lower curves. For example: \[Area=\int_{a}^{c} (f(x)-g(x))dx + \int_{c}^{b} (g(x)-f(x)) dx\] where \(f\) is greater than \(g\) over the interval \(x\in[a,c]\) and \(g\) is greater than \(f\) over the interval \(x\in[c,b]\).
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■ Note: due to the negative sign that is part of the area calculation, it will not matter if one or both of the curves is below the \(x\)-axis. |
15D.3 - Example 1: Determining the area between two curves
Find the area bounded by the curves \(f(x)=-x^2+2\) and \(g(x)=x^2\) between \(x=0\) and \(x=2\). The graphs of \(f(x)\) and \(g(x)\) are shown below.
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15D.3 - Example 1: Video Example
15D.3 - Example 1: Practice
Question 1 (TA):
Consider the curves \(f(x)=\frac{x^3}{8}-\frac{3x^2}{2}+\frac{9x}{2}\) and \(g(x)=-\frac{x^2}{8}+x\). Determine the area bounded by the graphs of \(f\) and \(g\) between \(x=0\) and \(x=7\), as shown below. Question 2:
Consider the functions \(f(x)=-x^2+2x+3\) and \(g(x)=x+1\).
Question 3 (TA):
Consider the function \(f(x)=x^3\).
15D.3 - Example 1: Solutions
Question 1: \(f(x)=g(x)\therefore x=0, x=4, x=7\) \[\therefore A=\int_{0}^{4} f(x)-g(x)dx+\int_{4}^{7} g(x)-f(x)dx\] \[\therefore A=\frac{937}{96}\] Question 2: (a) \(f(x)=g(x) \therefore x=-1, x=2\) (b) \(\int_{-1}^{2} f(x)-g(x) dx = \frac{9}{2}\) Question 3: (a) \(y=3x-2\) (b) \(x^3=3x-2 \therefore x=-2, x=1\) (c) \(\int_{-2}^{1} x^3-(3x-2)dx=\frac{27}{4}\) |
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Additional Exercises
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Topic Worksheets
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Other Resources
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Question 1 [IB, 2008]:
Find the area enclosed by the graphs of \(y=2+x-x^2\) and \(y=2-3x+x^2\). |
Solutions
Question 1: \[Area=\frac{8}{3}\] |
The area between two curves - Worksheet A