91Ó°ÊÓ

A wire filament is shaped to match the graph of the function of $y=\ln \left(2\cos x\right)$from $x=-\frac{\mathrm{Ï€}}{3}$to $x=\frac{\mathrm{Ï€}}{3}$, measured in inches, as shown in the figure. Given that the length of a curve y = f(x) from x = a to x = b can be calculated

with the formula${∫}_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx$find the length of the filament.

Suppose f(x) is continuous on R and that for some real number c, both

Consider the function $f\left(x\right)=\frac{\ln x}{x}$shown here:

(a) Find the average value of$f\left(x\right)$ on role="math" localid="1649276701979" $\left[\frac{1}{2},2\right]$.

(b) Find a value $c∈\left[\frac{1}{2},2\right]$ at which $f\left(x\right)$ achieves its average value.

Solve given definite integral.

${∫}_4^5 \frac{1}{x\sqrt{x^2+9}}dx$

Consider the function $f\left(x\right)=\ln x$.

(a) Find the signed area between the graph of $f\left(x\right)$and the $x$-axis on $\left[\frac{1}{2},4\right]$shown in the figure next at the left.

(b) Find the area between the graph of $f\left(x\right)$and the graph of $g\left(x\right)=1$on $\left[\frac{1}{2},4\right]$shown in the figure next at the right.

Prove the integration formula $∫secxdx=\ln |secx+\tan x|+C$

(a) by multiplying the integrand by a form of 1 and then applying a substitution;

(b) by differentiating ln | sec x + tan x|.

Consider the function $f\left(x\right)=\sin x\cos x$.

(a) Find the area between the graphs of $f\left(x\right)\mathrm{and}g\left(x\right)=\sin x$on localid="1649092422521" $[0,\mathrm{Ï€}]$shown next at the left.

(b) Find the area between the graphs of $f\left(x\right)\mathrm{and}h\left(x\right)=\sin 2x$on $[0,\mathrm{Ï€}]$shown next at the right.

Solve given definite integral.

${∫}_1^2 \frac{1}{x\sqrt{9−x^2}}dx$

Prove the integration formula $∫cscxdx=−\ln |cscx+cotx|+C$

(a) by multiplying the integrand by a form of 1 and then applying a substitution;

(b) by differentiating $−\ln |cscx+cotx|$.

Solve given definite integral.

${∫}_{1/4}^{1/2} \frac{1}{x^2\sqrt{1−x^2}}dx$