91Ó°ÊÓ

In Problems 3-6, calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. \(f(x)=4 x^{3}+1 ;[0,3]\) is divided into six equal subintervals, \(\bar{x}_{i}\) is the right end point.

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{4}(x+1)^{3 / 2} d x $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{3} \frac{2}{t^{3}} d t\)

Find the value of the indicated sum. $$ \sum_{k=3}^{7} \frac{(-1)^{k} 2^{k}}{(k+1)} $$

Find the value of the indicated sum. $$ \sum_{n=1}^{6} n \cos (n \pi) $$

In Problems 7-10, use the given values of a and b and express the given limit as a definite integral. \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\bar{x}_{i}\right)^{3} \Delta x_{i} ; a=1, b=3\)

In Problems 7-10, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16 .\) Present your approximations in a table like this: $$ \int_{1}^{3} \frac{1}{1+x^{2}} d x $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{0}^{4} \sqrt{t} d t\)

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{8} \sqrt[3]{w} d w\)

In Problems 7-10, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16 .\) Present your approximations in a table like this: $$ \int_{1}^{3} \frac{1}{x} d x $$