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Practice Problems
\(\textbf{1)}\) \(\displaystyle\int_{0}^{3}\left|x-2\right| \, dx\)
\(\textbf{2)}\) \(\displaystyle\int_{0}^{6}\left|2x-4\right| \, dx\)
\(\textbf{3)}\) \(\displaystyle\int_{0}^{4}\left|x+3\right| \, dx\)
\(\textbf{4)}\) \(\displaystyle\int_{0}^{4}\left|3x-6\right| \, dx\)
\(\textbf{5)}\) \(\displaystyle\int_{-4}^{4}\left|x\right| \, dx\)
\(\textbf{6)}\) \(\displaystyle\int_{0}^{3}\left|x+1\right| \, dx\)
\(\textbf{7)}\) \(\displaystyle\int_{-1}^{4}\left|3x-5\right| \, dx\)
\(\textbf{8)}\) \(\displaystyle\int_{-1}^{2}\left|5-x\right| \, dx\)
\(\textbf{9)}\) \(\displaystyle\int_{-7}^{-3}-\left|x+5\right| \, dx\)
See Related Pages\(\)
\(\bullet\text{ Absolute Value Integral Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
In Summary
Absolute value functions represent the distance between a number and zero on a number line. The notation for absolute value is is two vertical lines on either side of the number or variable, for example \(|x|\). The parent function \(f(x)=|x|\) consists of 2 rays that form a “V” shape with the vertex at the origin \((0,0)\) and the graph is \(y=x\) where \(x\gt0\) and \(y=-x\) where \(x\lt 0\). The vertex and slope of the lines can change with transformations. One variation of the equation with transformations would be \(f(x)=a|bx-h|+k\).
Integrals of absolute value functions involve calculating the area under the “V” shape. This is typically done by splitting the integral into two separate integrals split based on the x value of the the vertex. And then evaluating the integrals of each of the 2 line segments depending on the intervals of integration.
Integrals of absolute value functions are typically introduced in a calculus course, such as Calculus I or Calculus II. These concepts build upon foundational concepts learned in earlier math classes, such as algebra, trigonometry, and precalculus.
One common mistake that students often make is failing to properly set up the integral when solving a problem. If set up correctly you should be integrating over 2 line segments.
Five Real-World Examples of integrals
The distance traveled by a vehicle can be modeled by the integral of its velocity over time. If the velocity of the vehicle is given by a function v(t), then the distance traveled can be calculated as the integral of v(t) with respect to t.
The net change in the volume of a fluid flowing through a pipe can be calculated by the integral of the flow rate over time. If the flow rate of the fluid is given by a function f(t), then the net change in volume can be calculated as the integral of f(t) with respect to t.
The amount of heat transferred between two objects can be calculated by the integral of the heat transfer rate over time. If the heat transfer rate between the objects is given by a function h(t), then the amount of heat transferred can be calculated as the integral of h(t) with respect to t.
The total cost of producing a certain number of goods can be calculated by the integral of the cost per unit over the number of units produced. If the cost per unit is given by a function c(x), then the total cost can be calculated as the integral of c(x) with respect to x.
The total amount of a chemical substance absorbed by a living organism can be calculated by the integral of the absorption rate over time. If the absorption rate of the chemical is given by a function a(t), then the total amount absorbed can be calculated as the integral of a(t) with respect to t.
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