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Excel in math and science
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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
What is the surface area \(S\) of the body of revolution obtained by rotating the curve \(y=e^x,\) \(0 \le x \le 1,\) about the \(x-\)axis?
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 surface area \(S\) of the body of revolution obtained by rotating the parametric curve \[\begin{array} &x = 8 t^2 + 9 &y = -4 t &0 \leq t \leq 1 \end{array}\] about the \(x\)-axis?
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 surface area \(S\) of the solid of revolution obtained by rotating the parametric curve \[\begin{array} &x = 4 t &y = 6 t &0 \leq t \leq 2 \end{array}\] about the \(y\)-axis?
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 surface area \(S\) of the body of revolution obtained by rotating the parametric curve \[\begin{array} &x = 7 \cos t &y = 7 \sin t &0 \leq t \leq \pi \end{array}\] about the \(x\)-axis?
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 \(S\) is the surface area of the solid obtained by rotating the parametric curve \[\begin{array} &x = 4 \cos^3 t &y = 4 \sin^3 t &0 \leq t \leq \frac{\pi}{2} \end{array}\] about the \(x\)-axis, what is \(\frac{5}{2}S?\)