91Ó°ÊÓ

Consider the right triangle with vertices \((0,0),(0, b),\) and \((a, 0)\) where \(a>0\) and \(b>0 .\) Show that the average vertical distance from points on the \(x\) -axis to the hypotenuse is \(b / 2,\) for all \(a>0\).

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\pi / 3} \sec x \tan x d x$$

Average value of sine functions Use a graphing utility to verify that the functions \(f(x)=\sin k x\) have a period of \(2 \pi / k,\) where \(k=1,2,3, \ldots . .\) Equivalently, the first "hump" of \(f(x)=\sin k x\) occurs on the interval \([0, \pi / k] .\) Verify that the average value of the first hump of \(f(x)=\sin k x\) is independent of \(k .\) What is the average value?

Looking ahead: Integrals of \(\tan x\) and \(\cot x\) a. Use a change of variables to show that $$\int \tan x d x=-\ln |\cos x|+C=\ln |\sec x|+C$$ b. Show that $$\int \cot x d x=\ln |\sin x|+C$$.

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{8} \sqrt[3]{y} d y$$

Looking ahead: Integrals of sec \(x\) and \(\csc x\) a. Multiply the numerator and denominator of sec \(x\) by \(\sec x+\tan x ;\) then use a change of variables to show that $$\int \sec x d x=\ln |\sec x+\tan x|+C$$ b. Show that $$\int \csc x d x=-\ln |\csc x+\cot x|+C$$.

Equal areas The area of the shaded region under the curve \(y=2 \sin 2 x\) in (a) equals the area of the shaded region under the curve \(y=\sin x\) in (b). Explain why this is true without computing areas.

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{\sqrt{2}}^{2} \frac{d x}{x \sqrt{x^{2}-1}}$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2} \frac{z^{2}+4}{z} d z$$