91Ó°ÊÓ

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: $$ \int_{0}^{2} e^{-x^{2} / 2} d x $$

use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{-4}^{-2}\left(y^{2}+\frac{1}{y^{3}}\right) d y $$

Find the average value of the function on the given interval. $$ f(x)=e^{-x} ; \quad[0,2] $$

use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{4} \frac{s^{4}-8}{s^{2}} d s $$

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=4 .\) Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems. $$ \int_{0}^{2} 2 f(x) d x $$

Find the average value of the function on the given interval. $$ f(x)=\cosh (2 x) ; \quad[-2,2] $$

Use the given values of \(a\) and \(b\) and express the given limit as a definite integral. $$ \lim _{\|P\| \rightarrow 0} \sum_{i=1}^{n}\left(\sin \bar{x}_{i}\right)^{2} \Delta x_{i} ; a=0, b=\pi $$

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: $$ \int_{1}^{3} \ln \left(x^{2}+1\right) d x $$

Write the indicated sum in sigma notation. $$ 2+4+6+8+\cdots+50 $$

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=4 .\) Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems. $$ \int_{0}^{2}[2 f(x)+g(x)] d x $$