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Excel in math and science
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Area Between Curves
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
Let \(S\) be the area of the region bounded by the curve \(y=x^3\) and the tangent line to the curve at \((1,1)\). If \(S = \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, 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.
Let \(S_1\) be the area of the region bounded by \(y=x^2\) and \(y=1\). Let \(S_2\) be the area of the region bounded by \(y=x^2\) and \(y=\frac{7}{8} x^2+k,\) where \( k > 0 \). If \(S_2=S_1\), what is the value of \(\frac{1}{k}\)?
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 area of the region bounded by the \(x\)-axis and the graph of \(y=-k^{19} x^ {18} +k\), where \(k>0\). What is the value of \(152S\)?
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 \(S_1\) be the area of the region bounded by the curves \(y=x^2\) and \(y=1\). Let \(S_2\) be the area of the region bounded by the curves \(y=x^2\) and \[y=\frac{1}{m}x^2+\frac{m}{m-1},\] where \(m\) is a positive integer strictly greater than 1.
If \(a\) and \(b\) are coprime positive integers, and \(\frac{S_2}{S_1}=\frac{a}{b}\)and \(a+b=925\), what is \(m\)?
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 bounded by the parabola \( y=x(6-x)\) and the \(x\)-axis is divided into two regions by the line \(y=x\). If \(S_1\) is the region above \(y=x\) and \(S_2\) is the region below \(y=x\), then \(S_1 : S_2 = a : b\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?