Problem 72
Exact Trapezoid Rule Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.
Problem 72
Using the integral of sec \(^{3} u\) By reduction formula 4 in Section 8.3 $$ \int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C $$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\left(x^{2}-25\right)^{1 / 2},[5,10]$$
Problem 72
Area of a region between curves Find the area of the entire region bounded by the curves \(y=\frac{x^{3}}{x^{2}+1}\) and \(y=\frac{8 x}{x^{2}+1}\)
Problem 72
Between the sine and inverse sine Find the area of the region bounded by the curves \(y=\sin x\) and \(y=\sin ^{-1} x\) on the interval \([0,1 / 2]\).
Problem 72
Evaluate the following integrals. Assume a and b are real numbers and \(n\) is a positive integer. \(\int \frac{x}{\sqrt{a x+b}} d x \,\left(\text { Hint: } u^{2}=a x+b .\right)\)
Problem 72
A sine reduction formula Use integration by parts to obtain the reduction formula for positive integers \(n:\) $$ \int \sin ^{n} x d x=-\sin ^{n-1} x \cos x+(n-1) \int \sin ^{n-2} x \cos ^{2} x d x $$ Then use an identity to obtain the reduction formula $$ \int \sin ^{n} x d x=-\frac{\sin ^{n-1} x \cos x}{n}+\frac{n-1}{n} \int \sin ^{n-2} x d $$ Use this reduction formula to evaluate \(\int \sin ^{6} x d x\)
Problem 72
Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(x\) -axis on the interval (-1,1] is revolved about the line \(y=-1\).
Problem 73
Evaluate the following integrals. Assume a and b are real numbers and \(n\) is a positive integer. \(\int x(a x+b)^{n} d x(\text { Hint: } u=a x+b .)\)
Problem 73
Two useful exponential integrals Use integration by parts to derive the following formulas for real numbers \(a\) and \(b\). $$\begin{array}{l} \int e^{a x} \sin b x d x=\frac{e^{a x}(a \sin b x-b \cos b x)}{a^{2}+b^{2}}+C \\\ \int e^{a x} \cos b x d x=\frac{e^{a x}(a \cos b x+b \sin b x)}{a^{2}+b^{2}}+C \end{array}$$
Problem 73
Using the integral of sec \(^{3} u\) By reduction formula 4 in Section 8.3 $$ \int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C $$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\sqrt{x^{2}-9} / x,[3,6]$$
