Problem 38
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{0}^{\pi / 3} 4 \sec x \tan x d x$$
Problem 38
Consider the integral \(\int_{-1}^{1} \sin \left(x^{2}\right) d x\) $$\begin{array}{l}{\text { (a) Find } f^{(4)} \text { for } f(x)=\sin \left(x^{2}\right). \text { (You may want to check your work with a CAS if you have one available.) }}\end{array} $$ (b) Graph \(y=f^{(4)}(x)\) in the viewing window \([-1,1]\) by \([-30,10] .\) (c) Explain why the graph in part (b) suggests that \(\left|f^{(4)}(x)\right| \leq 30\) for \(-1 \leq x \leq 1\) (d) Show that the error estimate for Simpson's Rule in this case becomes $$\left|E_{S}\right| \leq \frac{h^{4}}{3}$$ (e) Show that the Simpson's Rule error will be less than or equal to 0.01 if \(h \leq 0.4 .\) (f) How large must \(n\) be for \(h \leq 0.4 ?\)
Problem 38
Rectangular Approximation Methods Prove or disprove the following statement: MRAM \(_{n}\) is always the average of LRAM \(_{n}\) and \(\operatorname{RRAM}_{n}\).
Problem 39
In Exercises \(37-40,\) (a) find the points of discontinuity of the integrand on the interval of integration, and (b) use area to evaluate the integral. $$\int_{-3}^{4} \frac{x^{2}-1}{x+1} d x$$
Problem 39
Rectangular Approximation Methods Show that if \(f\) is a nonnegative function on the interval \([a, b]\) and the line \(x=(a+b) / 2\) is a line of symmetry of the graph of \(y=f(x)\) then \(L R A M_{n} f=\operatorname{RRAM}_{n} f\) for every positive integer \(n .\)
Problem 39
Writing to Learn If \(a v ( f )\) really is a typical value of the integrable function \(f ( x )\) on \([ a , b ]\) , then the number \(a v ( f )\) should have the same integral over \([ a , b ]\) that \(f\) does. Does it? That is, does \(\int _ { a } ^ { b } a v ( f ) d x = \int _ { a } ^ { b } f ( x ) d x ?\) Give reasons for your answer.
Problem 39
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{-1}^{1}(r+1)^{2} d r$$
Problem 40
(Continuation of Exercise 37\()\) (a) Inscribe a regular \(n\) -sided polygon inside a circle of radius 1 and compute the area of one of the \(n\) congruent triangles formed by drawing radii to the vertices of the polygon. (b) Compute the limit of the area of the inscribed polygon as \(n \rightarrow \infty\) (c) Repeat the computations in parts (a) and (b) for a circle of radius \(r .\)
Problem 40
In Exercises \(37-40,\) (a) find the points of discontinuity of the integrand on the interval of integration, and (b) use area to evaluate the integral. $$\int_{-5}^{6} \frac{9-x^{2}}{x-3} d x$$
Problem 40
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{0}^{4} \frac{1-\sqrt{u}}{\sqrt{u}} d u$$
