91Ó°ÊÓ

Evaluate the limit by expressing it as a definite integral over the interval \([a, b]\) and applying appropriate formulas from geometry. $$\lim _{{\max }{\Delta x_{k} \rightarrow 0}}{\sum_{k=1}^{n} \sqrt{4-\left(x_{k}^{*}\right)^{2}}} \Delta x_{k} ; a=-2, b=2$$

Evaluate the integrals by any method. $$\int_{\pi^{2}}^{4 \pi^{2}} \frac{1}{\sqrt{x}} \sin \sqrt{x} d x$$

Determine whether the statement is true or false. Explain your answer. Every integral curve of the slope field $$\frac{d y}{d x}=\frac{1}{\sqrt{x^{2}+1}}$$ is the graph of an increasing function of \(x\).

Evaluate the integrals using appropriate substitutions. $$\int x \sec ^{2}\left(x^{2}\right) d x$$

Use a graphing utility to generate some representative integral curves of the function \(f(x)=5 x^{4}-\sec ^{2} x\) over the interval \((-\pi / 2, \pi / 2)\)

Use a calculating utility to find the midpoint approximation of the integral using \(n=20\) sub-intervals, and then find the exact value of the integral using Part 1 of the Fundamental Theorem of Calculus. $$\int_{1}^{3} \frac{1}{x^{2}} d x$$

Evaluate the integrals by any method. $$\int_{\pi / 12}^{\pi / 9} \sec ^{2} 3 \theta d \theta$$

Evaluate the integrals using appropriate substitutions. $$\int \frac{\cos 4 \theta}{(1+2 \sin 4 \theta)^{4}} d \theta$$

Evaluate the integrals by any method. $$\int_{0}^{\pi / 6} \tan 2 \theta d \theta$$

Use a graphing utility to generate some representative integral curves of the function \(f(x)=(x-1) / x\) over the interval (0,5)