Problem 3
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. $$ \int_{0}^{2} \sqrt{x} d x $$
Problem 3
Find the value of the indicated sum. $$ \sum_{k=1}^{7} \frac{1}{k+1} $$
Problem 4
Find the average value of the function on the given interval. $$ f(x)=\frac{x^{2}}{\sqrt{x^{3}+16}} ; \quad[0,2] $$
Problem 4
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. $$ \int_{1}^{3} x \sqrt{x^{2}+1} d x $$
Problem 4
use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{2}\left(4 x^{3}+7\right) d x $$
Problem 4
Find the value of the indicated sum. $$ \sum_{l=3}^{8}(l+1)^{2} $$
Problem 4
Calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. $$ \begin{array}{r} \text f(x)=-x / 2+3 ; P:-3<-1.3<0<0.9<2 ; \\ \bar{x}_{1}=-2, \bar{x}_{2}=-0.5, \bar{x}_{3}=0, \bar{x}_{4}=2 \end{array} $$
Problem 5
Calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. \(f(x)=x^{2} / 2+x ;[-2,2]\) is divided into eight equal subintervals, \(\bar{x}_{i}\) is the midpoint.
Problem 5
use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{4} \frac{1}{w^{2}} d w $$
Problem 5
Find the value of the indicated sum. $$ \sum_{m=1}^{8}(-1)^{m} 2^{m-2} $$
