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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.
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including olympiad champions, researchers, and professionals.
Arc Length and Surface Area
Finding the perimeter of arbitrary curves and the area of 3D shapes foils the traditional tools of Geometry, and calls for the help of integrals and derivatives to make these calculations.
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 area of the surface obtained by revolving the curve \( y = \sqrt{x} \), \( 4 \leq x \leq 7 \), about the \( x \)-axis can be represented by \( \frac{\pi}{6}(a \sqrt{a} - b \sqrt{b}) \). What is the value of \( a + b \)?
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.
The area of the plane \( 3x+2y+z = 30 \), where \(x \geq 0\), \(y \geq 0\) and \(z \geq 0\) can be represented by \(a \sqrt{b}\), where \(a\) and \(b\) are positive integers and \(b\) is not divisible by the square of a prime. What is the value of \(a+b\)?
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 area of the surface obtained by revolving the curve \(y=\sqrt{81-x^2} \) from \( -2 \leq x \leq 2\) 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.
Let \(S\) be the surface area of the solid obtained by rotating the curve \( y = x^3 \) \(( 0 \leq x \leq 1) \) about the \( x \)-axis. If \(S = \frac{\pi}{27}(a \sqrt{a} - 1) \), where \(a\) is a positive integer, what is the value of \( a \)?
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.
The area of the paraboloid \( z = x^2 + y^2 \), where \( 0 \leq z \leq 121 \), can be represented by \( \frac{\pi}{6} (a \sqrt{a} - 1) \). What is the value of \(a\)?