Notes
Problems
Find the arc length over the given interval.
(Calculator needed for integrals)
\(\textbf{1)}\) \( f(x)=4(x+1)^{3/2} \,\,\,\, [2,5] \)
\(\textbf{2)}\) \( y=\sin x \,\,\,\, [0,\pi] \)
\(\textbf{3)}\) \( y=x^2 \,\,\,\, [0,4] \)
\(\textbf{4)}\) \( y=\frac{\sqrt[3]{x}}{2} \,\,\,\, [1,3] \)
\(\textbf{5)}\) \( y=\frac{1}{x} \,\,\,\, [1,5] \)
\(\textbf{6)}\) \( y=\frac{1}{x} \,\,\,\, [.1,1] \)
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\) \(\text{Arc Length Calculator (Wolfram Alpha)}\)
\(\bullet\) \(\text{Integral Calculator (Wolfram Alpha)}\)
\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\text{Volume}=\displaystyle \int_{a}^{b}\left(\text{Area}\right) \, dx…\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
In Summary
In calculus, arc length refers to the distance along a curved path. It is a measure of the distance between two points along a curve, rather than a straight line. Arc length is an important concept in calculus because it allows us to measure and understand the properties of curved objects and shapes. It is an advanced topic that builds upon the concepts of limits and derivatives, which are introduced in a first-year calculus course.
Arc length is related to several other concepts in calculus, including integration, limits, and derivatives. It is also closely related to the concept of curvature. Other related topics include circular arc length, the circumference of a circle, and the arc length of a curve that is not straight. These concepts are important in fields such as physics, engineering, and geometry.”