Problem 73
Area of a region between curves Find the area of the region bounded by the curves \(y=\frac{x^{2}}{x^{3}-3 x}\) and \(y=\frac{1}{x^{3}-3 x}\) on the interval [2,4]
Problem 73
Are length of an ellipse The length of an ellipse with axes of length \(2 a\) and \(2 b\) is $$ \int_{0}^{2 \pi} \sqrt{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} d t $$ Use numerical integration, and experiment with different values of \(n\) to approximate the length of an ellipse with \(a=4\) and \(b=8\)
Problem 73
Evaluate the following integrals. $$\int_{9}^{16} \sqrt{1+\sqrt{x}} d x$$
Problem 73
Evaluate \(\int \frac{d x}{x^{2}-1},\) for \(x>1,\) in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.
Problem 73
A tangent reduction formula Prove that for positive integers \(n \neq 1\) $$ \int \tan ^{n} x d x=\frac{\tan ^{n-1} x}{n-1}-\int \tan ^{n-2} x d x $$ Use the formula to evaluate \(\int_{0}^{x / 4} \tan ^{3} x d x\)
Problem 74
Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=-\ln x\) and the \(x\) -axis on the interval (0,1] is revolved about the \(x\) -axis.
Problem 74
sine integral The theory of diffraction produces the sine integral function \(\mathrm{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t .\) Use the Midpoint Rule to approximate \(\left.\operatorname{Si}(1) \text { and } \operatorname{Si}(10) . \text { (Recall that } \lim _{x \rightarrow 0} \frac{\sin x}{x}=1 .\right)\) Experiment with the number of subintervals until you obtain approximations that have an error less than \(10^{-3}\). A rule of thumb is that if two successive approximations differ by less than \(10^{-3}\), then the error is usually less than \(10^{-3} .\)
Problem 74
Volume of a solid Consider the region \(R\) bounded by the graph of \(f(x)=\frac{1}{x+2}\) and the \(x\) -axis on the interval \([0,3] .\) Find the volume of the solid formed when \(R\) is revolved about the \(y\) -axis.
Problem 74
Evaluate the following integrals. Assume a and b are real numbers and \(n\) is a positive integer. \(\int x^{n} \sin ^{-1} x d x\) (Hint. integration by parts.)$
Problem 74
A secant reduction formula Prove that for positive integers \(n \neq 1\) $$ \int \sec ^{n} x d x=\frac{\sec ^{n-2} x \tan x}{n-1}+\frac{n-2}{n-1} \int \sec ^{n-2} x d x $$ (Hint: Integrate by parts with \(u=\sec ^{n-2} x\) and \(d v=\sec ^{2} x d x\) )
