Problem 75
Find the area of the region bounded by the curve \(f(x)=\left(16+x^{2}\right)^{-3 / 2}\) and the \(x\) -axis on the interval [0,3]
Problem 76
Shortcut for the Trapezoid Rule Given a Midpoint Rule approximation \(M(n)\) and a Trapezoid Rule approximation \(T(n)\) for a continuous function on \([a, b]\) with \(n\) subintervals, show that \(T(2 n)=\frac{T(n)+M(n)}{2}\)
Problem 76
Different substitutions a. Show that \(\int \frac{d x}{\sqrt{x-x^{2}}}=\sin ^{-1}(2 x-1)+C\) using cither \(u=2 x-1\) or \(u=x-\frac{1}{2}\) b. Show that \(\int \frac{d x}{\sqrt{x-x^{2}}}=2 \sin ^{-1} \sqrt{x}+C\) using \(u=\sqrt{x}\) c. Prove the identity \(2 \sin ^{-1} \sqrt{x}-\sin ^{-1}(2 x-1)=\frac{\pi}{2}\)
Problem 76
Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{\cos \theta}{\left(\sin ^{3} \theta-4 \sin \theta\right)} d \theta$$
Problem 76
Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=\left(x^{2}-1\right)^{-1 / 4}\) and the \(x\) -axis on the interval (1,2] is revolved about the \(y\) -axis.
Problem 76
Evaluate the following integrals. $$\int \frac{x}{x^{2}+6 x+18} d x$$
Problem 76
Find the error Suppose you evaluate \(\int \frac{d x}{x}\) using integration by parts. With \(u=1 / x\) and \(d v=d x,\) you find that \(d u=-1 / x^{2} d x\) \(v=x,\) and $$\int \frac{d x}{x}=\left(\frac{1}{x}\right) x-\int x\left(-\frac{1}{x^{2}}\right) d x=1+\int \frac{d x}{x}$$ You conclude that \(0=1 .\) Explain the problem with the calculation.
Problem 77
Surface area Let \(f(x)=\sqrt{x+1}\). Find the area of the surface generated when the region bounded by the graph of \(f\) and the \(x\) -axis on the interval [0,1] is revolved about the \(x\) -axis.
Problem 77
Are length of a parabola Find the length of the curve \(y=a x^{2}\) from \(x=0\) to \(x=10,\) where \(a > 0\) is a real number.
Problem 77
Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{e^{x}}{\left(e^{x}-1\right)\left(e^{x}+2\right)} d x$$
