1. The work done by a force F acting along a material path L can be calculated using the integral A = ∫_0^S F(s) ds, where s is the parameter of the path.
2. The mass of an object with non-uniform density ρ(x) along the x-axis between a and b can be found using the integral m = ∫_a^b ρ(x) dx.
3. The coordinates (xc, yc) of the centroid of a curve given in parametric form x=φ(t), y=ψ(t) between α and β can be determined using the integrals xc = ∫_α^
The document provides information about Riemann sums and Riemann integrals. It states that if f(x) is a non-negative function on the interval [a,b], then the graph of y=f(x) can be approximated by dividing the interval into n subintervals and taking the left or right Riemann sum of the resulting rectangles. The Riemann integral from a to b of f(x) is defined as the limit of these Riemann sums as n approaches infinity. Newton and Leibniz developed the fundamental theorem of calculus that relates integrals to derivatives.
1. The document is an English language learning exercise that presents sentences in Mongolian with blanks to be filled in with English verbs. It covers topics like present continuous tenses, verb phrases, and filling in missing verbs.
2. The exercises include choosing the correct verbs to fill in blanks and match verb phrases with situations. Examples ask about what noises people are hearing, what friends are doing on holiday, and whether people enjoy their jobs or classes.
3. The purpose is to practice English verb tenses and verb phrases in sentences about common situations to build English language skills.
1. The work done by a force F acting along a material path L can be calculated using the integral A = ∫_0^S F(s) ds, where s is the parameter of the path.
2. The mass of an object with non-uniform density ρ(x) along the x-axis between a and b can be found using the integral m = ∫_a^b ρ(x) dx.
3. The coordinates (xc, yc) of the centroid of a curve given in parametric form x=φ(t), y=ψ(t) between α and β can be determined using the integrals xc = ∫_α^
The document provides information about Riemann sums and Riemann integrals. It states that if f(x) is a non-negative function on the interval [a,b], then the graph of y=f(x) can be approximated by dividing the interval into n subintervals and taking the left or right Riemann sum of the resulting rectangles. The Riemann integral from a to b of f(x) is defined as the limit of these Riemann sums as n approaches infinity. Newton and Leibniz developed the fundamental theorem of calculus that relates integrals to derivatives.
1. The document is an English language learning exercise that presents sentences in Mongolian with blanks to be filled in with English verbs. It covers topics like present continuous tenses, verb phrases, and filling in missing verbs.
2. The exercises include choosing the correct verbs to fill in blanks and match verb phrases with situations. Examples ask about what noises people are hearing, what friends are doing on holiday, and whether people enjoy their jobs or classes.
3. The purpose is to practice English verb tenses and verb phrases in sentences about common situations to build English language skills.