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Excel in math and science
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It's hard to learn from lectures and videos
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Calculus of Parametric Equations
Parametric equations can be quite handy, and we don't want to unravel them just to do Calculus. Some tricks can bend traditional derivative and integral methods to apply to parametric equations.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
Challenge Quizzes
The location of a dot \(P\) at a given time \(t\) in the \(xy\) plane is given by \((x,y) = (t - \sin t, 1 - \cos t)\). What is the distance traveled by \(P\) in the interval \(0 \leq t \leq 2\pi\)?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
What is the length of the curve parametrized by the equations \[\begin{array} \displaystyle x=e^{2t}\cos t, & y=e^{2t}\sin t,\end{array}\] in the domain \(0 \leq t \leq 4 ?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If \(x=4\sin^2 t\) and \(y=4\cos^2 t,\) what is the distance traveled by the point \(P=(x,y)\) during the time interval \(0 \leq t \leq 5\pi?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Given the curve \( H(t) = \frac{2}{3} (t+4)^{3/2} \), the arc length of the graph between \(t=4\) and \(t=12\) can be expressed in the form \[ \frac {a\sqrt{b}}{c} - d \] where \(a\), \(b\), \(c\), and \(d\) are positive integers, \(a\) and \(c\) are coprime, and \(b\) is not divisible by the square of any prime. What is \(a+b+c+d\)?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Given the curve defined by \(x = t^3\) and \(y = t^2,\) what is the length of the curve from \(t=0\) to \(t= 10?\)