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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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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
Suppose \(T, L,\) and \(R\) are the approximation values using the Trapezium rule, left Riemann sum, and right Riemann sum, respectively, of the integral \(\displaystyle{\int_0^{1}(16-x^2)dx}\) using \(19\) intervals. Which of the following is the correct relationship between the three approximations?
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.
Suppose \(T, L,\) and \(R\) are the approximation values using the Trapezium rule, left Riemann sum, and right Riemann sum, respectively, of the integral \(\displaystyle{\int_0^{17}(2x^3+3)dx}\) using \(15\) intervals. Which of the following is the correct relationship between the three approximation values?
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.
Which method gives the closest approximate value to the integral \(\displaystyle{\int_0^{4}4x^2 dx},\) a right Riemann sum, a left Riemann sum, or the Trapezium rule, where the same number of intervals is used for each 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.
Suppose the integral \(\displaystyle{\int_0^{3}3x^{3}dx}\) is approximated using a right Riemann sum and a left Riemann sum. Which sum gives a more accurate approximation to the integral?
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.
Sarah is approximating the integral \(\displaystyle{\int_0^{3}(4x+2)dx}\) using a Riemann sum. Would a right Riemann sum or left Riemann sum give a more accurate approximation?