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Excel in math and science
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Numerical Approximation of Integrals
Whether you want to program a calculator or just approximation an integral on paper, break out some numerical approximation tools like Simpson's Rule and the Trapezoid Rule.
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
Robert is approximating the integral \(\displaystyle{\int_0^{2}8x^3dx}\) using a right Riemann sum. If Robert takes the limit of the right Riemann sum with infinitely many intervals, what is the resulting approximation?
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.
Stephanie is approximating the integral \(\displaystyle{\int_0^1 (2-x^2) dx}\) on \(3\) intervals using either a Riemann sum or the Trapezium rule. What would happen if she uses \(7\) intervals instead of \(3?\)
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
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Max is applying the Trapezium rule to approximate \(\displaystyle{\int_0^{\frac{\pi}{2}} 2\sin x dx}\) using \(9\) intervals. What would happen if he uses \(5\) intervals instead of \(9?\)
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
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including olympiad champions, researchers, and professionals.
In using a Riemann sum to approximate an integral, the larger the number of intervals, the more accurate the approximation. If we are approximating \(\displaystyle{\int_0^1(x^2+2)dx}\) using a left Riemann sum, what is the minimum number of intervals needed in order to make the error less than \(0.1?\)
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.
Rebecca is approximating the integral \(\displaystyle{\int_0^{4}(6x^2+2)dx}\) using the Trapezium rule. If Rebecca takes the limit of the number of intervals to infinity, what is the approximation she obtains?