91Ó°ÊÓ

Let \(A\) be the area under the graph of the given function \(f\) from \(a\) to \(b\). Approximate \(A\) by dividing \([a, b]\) into subintervals of equal length \(\Delta x\) and using (a) \(A_{1 P}\) and \((b) A_{C P}\) \(f(x)=x^{2}+1 ; \quad a=1, \quad b=3 ; \quad \Delta x=\frac{1}{2}\)

Exer. 9-48: Evaluate the integral. $$ \int\left(1+\frac{1}{x}\right)^{-3}\left(\frac{1}{x^{2}}\right) d x $$

Express as one integral. $$ \int_{c}^{m} f(x) d x-\int_{d}^{m} f(x) d x $$

Suppose the table of values for \(x\) and \(y\) was obtained empirically. Assuming that \(y=f(x)\) and \(f\) is continuous, approximate \(\int_{2}^{4} f(x) d x\) by means of \(\mid\) a) the trapezoidal rule and (b) Simpson's rule. $$ \begin{array}{|c|c|c|c|} \hline x & 2.0 & 3.0 & 4.0 \\ \hline y & 5 & 4 & 3 \\ \hline \end{array} $$

Let \(A\) be the area under the graph of the given function \(f\) from \(a\) to \(b\). Approximate \(A\) by dividing \([a, b]\) into subintervals of equal length \(\Delta x\) and using (a) \(A_{1 P}\) and \((b) A_{C P}\) \(f(x)=4-x^{2} ; \quad a=0, \quad b=2 ; \quad \Delta x=\frac{1}{2}\)

Given \(\int_{1}^{4} \sqrt{x} d x=\frac{14}{3},\) evaluate the integral. $$ \int_{4}^{4} \sqrt{x} d x $$

Evaluate. $$ \int-\frac{1}{5} \sin u d u $$

Evaluate. $$ \int_{1}^{2} \frac{x^{2}+2}{x^{2}} d x $$

Evaluate the integral. $$ \int_{-1}^{5}|2 x-3| d x $$

Evaluate. $$ \int \frac{7}{\csc x} d x $$