Problem 70
Different methods Let \(I=\int \frac{x+2}{x+4} d x\) a. Evaluate \(I\) after first performing long division on the integrand. b. Evaluate \(I\) without performing long division on the integrand. c. Reconcile the results in parts (a) and (b).
Problem 70
67-70. Integrals of the form \(\int \sin m x\) cos nux dx Use the following three identities to evaluate the given integrals. $$ \begin{aligned} \sin m x \sin n x &=\frac{1}{2}(\cos ((m-n) x)-\cos ((m+n) x)) \\ \sin m x \cos n x &=\frac{1}{2}(\sin ((m-n) x)+\sin ((m+n) x)) \\\ \cos m x \cos n x &=\frac{1}{2}(\cos ((m-n) x)+\cos ((m+n) x)) \end{aligned} $$ $$\int \cos x \cos 2 x \, d x$$
Problem 70
Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=e^{-x^{2}}\) a. Find a Simpson's Rule approximation to \(\int_{0}^{3} e^{-x^{2}} d x\) using \(n=30\) subintervals.b. Calculate \(f^{(4)}(x)\) c. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1. (Hint: Use a graph to find an upper bound for \(\left.\left|f^{(4)}(x)\right| \text { on }[0,3] .\right)\)
Problem 71
Solid of revolution Find the volume of the solid generated when the region bounded by \(y=\cos x\) and the \(x\) -axis on the interval \([0, \pi / 2]\) is revolved about the \(y\) -axis.
Problem 71
Evaluate the following integrals. Assume a and b are real numbers and \(n\) is a positive integer. \(\int \frac{x}{a x+b} d x\) (Hint: \,\(u=a x+b\).)
Problem 71
Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=\sqrt{\sin x}\) a. Find a Simpson's Rule approximation to \(\int_{1}^{2} \sqrt{\sin x} d x\) using \(n=20\) subintervals. b. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1. (Hint: Use the fact that \(\left.\left|f^{(4)}(x)\right| \leq 1 \text { on }[1,2] .\right)\)
Problem 71
Evaluate the following integrals. $$\int \frac{1-\cos x}{1+\cos x} d x$$
Problem 71
Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=(x-1)^{-1 / 4}\) and the \(x\) -axis on the interval (1,2] is revolved about the \(x\) -axis.
Problem 71
Different methods Let \(I=\int \frac{x^{2}}{x+1} d x\) a. Evaluate \(I\) using the substitution \(u=x+1\) b. Evaluate \(I\) after first performing long division on the integrand. c. Reconcile the results in parts (a) and (b).
Problem 71
Prove the following orthogonality relations (which are used to generate Fourier series). Assume \(m\) and \(n\) are integers with \(m \neq n\) a. \(\int_{0}^{\pi} \sin m x \sin n x d x=0\) b. \(\int_{0}^{\pi} \cos m x \cos n x d x=0\) c. \(\int_{0}^{\pi} \sin m x \cos n x d x=0,\) for \(|m+n|\) even
