Solution Found!
Solution: Care with the secant substitution Recall that the
Chapter 4, Problem 78AE(choose chapter or problem)
Care with the secant substitution Recall that the substitution \(x=a \sec \theta\) implies that \(x \geq a\) (in which case \(0 \leq \theta<\pi / 2\) and \(\tan \theta \geq 0\) ) or \(x \leq-a\) (in which case \(\pi / 2<\theta \leq \pi\) and \(\tan \theta \leq 0\) ).
Graph the function \(f(x)=\frac{1}{x \sqrt{x^{2}-36}}\) on its domain. Then, find the area of the region \(R_{1}\) bounded by the curve and the x-axis on \([-12,-12 / \sqrt{3}]\) and the region \(R_{2}\) bounded by the curve and the x-axis on \([12 / \sqrt{3}, 12]\). Be sure your results are consistent with the graph.
Questions & Answers
QUESTION:
Care with the secant substitution Recall that the substitution \(x=a \sec \theta\) implies that \(x \geq a\) (in which case \(0 \leq \theta<\pi / 2\) and \(\tan \theta \geq 0\) ) or \(x \leq-a\) (in which case \(\pi / 2<\theta \leq \pi\) and \(\tan \theta \leq 0\) ).
Graph the function \(f(x)=\frac{1}{x \sqrt{x^{2}-36}}\) on its domain. Then, find the area of the region \(R_{1}\) bounded by the curve and the x-axis on \([-12,-12 / \sqrt{3}]\) and the region \(R_{2}\) bounded by the curve and the x-axis on \([12 / \sqrt{3}, 12]\). Be sure your results are consistent with the graph.
ANSWER:Problem 78AE
Care with the secant substitution Recall that the substitution x = a sec θ implies that x ≥ a (in which case 0 ≤ θ<π/2 and tan θ ≥ 0) or x ≤ –a (in which case π/2 < θ ≤ π and tan θ ≤ 0).
Graph the function on its domain. Then, find the area of the region R1 bounded by the curve and the x-axis on [–12, ] and the region R2 bounded by the curve and the x-axis on [, 12]. Be sure your results are consistent with the graph.
SOLUTION
Step 1
We have to find the graph of the function
