15D.2 - Area under a curve
We will examine three cases for the area under a curve:
Areas above the \(x\)-axis:
For a function \(f\) which is above the \(x\)-axis over the interval \(x\in [a,b]\), where \(b>a\). The area is simply given by: \[\int_{a}^{b} f(x) dx\]
15D.2 - Example 1: Finding the area under a curve
Find the area under the curve \(y=x^3+2\) between \(x=-1\) and \(x=1\). The graph of the curve is shown below.
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15D.2 - Example 1: Video Example
15D.2 - Example 1: Practice
Question 1: Find the area under the graph of \(y=x^2\) between \(x=-1\) and \(x=2\). Question 2: Find the area bounded by the parabola \(y=-x^2+2x+3\) and the \(x\)-axis between the axial intercepts of the graph. Question 3: Find the area under the graph of \(y=-x^2+3\) between \(x=0\) and \(x=\sqrt{3}\). Question 4: The area under the curve of \(y=3x^2\) between \(x=-a\) and \(x=a\), where \(a\in R^+\), is \(16\) square units. Find the value of \(a\). 15D.2 - Example 1: Solutions
Question 1: \[A=\int_{-1}^{2} x^2dx=3\] Question 2: \[A=\int_{-1}^{3}-x^2+2x+3dx=\frac{32}{3}\] Question 3: \[A=\int_{0}^{\sqrt{3}} -x^2+3dx=2\sqrt{3}\] Question 4: \[\int_{-a}^{a}3x^2dx=16 \therefore a=2\] |
15D.2 - Example 2: Finding the area under a curve
Find the area under the curve \(y=x^3+3x^2-6x+4\) between \(x=-2\) and \(x=1\). The graph of the curve is shown below.
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15D.2 - Example 2: Video Example
15D.2 - Example 2: Practice
Question 1: Find the area under the curve of \(y=-x^3+4x^2\) between \(x=0\) and \(x=4\). Question 2 (TA): Find the value of \(a\), where \(a>-2\) such that the area under the curve of \(f(x)=x^2+1\) between \(x=-2\) and \(x=a\) is \(30\) square units. 15D.2 - Example 2: Solutions
Question 1: \[A=\int_{0}^{4}-x^3+4x^2dx=\frac{64}{3}\] Question 2: \[\int_{-2}^{a}x^2+1dx=30\] \[\therefore a=4\] |
Areas below the \(x\)-axis:
For a function \(f\) which is below the \(x\)-axis over the interval \(x\in [a,b]\), where \(b>a\). The calculation \(\int_{a}^{b} f(x) dx\) will return a negative value; however, area is a physical quantity and cannot be negative. Therefore, we need to correct the value to make it positive when finding the area. This can be done several ways:
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15D.2 - Example 3: Finding the area under a curve
Find the area between the curve \(y=x^2-16\) and the \(x\)-axis between \(x=-4\) and \(x=4\). The graph of the curve is shown below.
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15D.2 - Example 3: Video Example
15D.2 - Example 3: Practice
Question 1: Find the area bounded by the function \(f(x)=x^3\) and the \(x\)-axis between \(x=-3\) and \(x=-1\). Question 2: Find the area between the curve of \(y=x^4-1\) and the \(x\)-axis between \(x=-1\) and \(x=1\). Question 3: Find the area bounded by \(y=4\sqrt[3]{x}\) and the \(x\)-axis between \(x=-1\) and \(x=0\). 15D.2 - Example 3: Solutions
Question 1: \[A=-\int_{-3}^{-1} x^3dx=20\] Question 2: \[A=-\int_{-1}^{1} x^4-1dx=\frac{8}{5}\] Question 3: \[A=-\int_{-1}^{0}4\sqrt[3]{x}dx=3\] |
15D.2 - Example 4: Finding the area under a curve
Find the area between the curve \(y=x^3-x^2-5x-3\) and the \(x\)-axis between \(x=-1\) and \(x=3\). The graph of the curve is shown below.
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15D.2 - Example 4: Video Example
15D.2 - Example 4: Practice
Question 1: Find the area fully enclosed by \(y=x^4-4x^3\) and the \(x\)-axis. Question 2: Find the value of \(a\) such that the area bounded by the function \(g(x)=a(x^2-4)\) between \(x=-2\) and \(x=2\) is \(32\) square units. 15D.2 - Example 4: Solutions
Question 1: \[A=-\int_{0}^{4} x^4-4x^3dx=\frac{256}{5}\] Question 2: \[a\int_{-2}^{2} x^2-4=32\therefore a=3\] |
Mixed areas:
A mixed area is one where the curve is cuts the \(x\)-axis at some point during the region of integration. Therefore, some of the area is below the \(x\)-axis ("negative") and some of the area is above the \(x\)-axis ("positive"). When determining the area between the curve and the \(x\)-axis, you must split the definite integral up into regions that lay above the \(x\)-axis and areas that lay below the \(x\)-axis.
- For the regions above the \(x\)-axis calculate the area directly using the fundamental theorem of calculus.
- For the regions below the \(x\)-axis you must make the area "positive" by placing a negative sign in front of the integral or by swapping the terminals of integration.
15D.2 - Example 5: Finding the area under a curve
Find the area bounded by the curve \(f(x)=3x^2-3\) and the \(x\)-axis over the interval \(x\in [0,2]\). The graph of the curve is shown below.
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15D.2 - Example 5: Video Example
15D.2 - Example 5: Practice
15D.2 - Example 5: Solutions
Question 1: \[A=\int_{0}^{2}f(x)dx-\int_{2}^{4}f(x)dx=8\] Question 2: \[A=-\int_{-2}^{0} g(x)dx+\int_{0}^{1}g(x)dx = \frac{29}{5}\] |
15D.2 - Example 6: Finding the area under a curve (CAS)
Find the area bounded by the curve \(f(x)=(x+2)(x+1)(x-2)\) and the \(x\)-axis over the interval \(x\in [-2,2]\). The graph of the curve is shown below.
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15D.2 - Example 6: Video Example
15D.2 - Example 6: Practice
15D.2 - Example 6: Solutions
Question 1: \[A=\int_{-2}^{1}f(x)dx-\int_{1}^{3}f(x)dx=\frac{243}{12}\] Question 2: \[\int_{0}^{c}x^2-1dx=0\therefore c=\sqrt{3}\] |