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Chapter 5: Problem 60
Find the value of \(c\) such that the region bounded by \(y=c \sin x\) and the \(x\) -axis on the interval \([0, \pi]\) has area 1.
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Suppose \(f\) is continuous on \([0, \infty)\) and \(A(x)\) is the net area of the region bounded by the graph of \(f\) and the \(t\) -axis on \([0, x] .\) Show that the maxima and minima of \(A\) occur at the zeros of \(f\). Verify this fact with the function \(f(x)=x^{2}-10 x\)
Consider the integral \(I(p)=\int_{0}^{1} x^{p} d x\) where \(p\) is a positive integer. a. Write the left Riemann sum for the integral with \(n\) subintervals. b. It is a fact (proved by the 17 th-century mathematicians Fermat and Pascal) that \(\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{p}=\frac{1}{p+1} \cdot\) Use this fact to evaluate \(I(p)\)
Consider the function g. which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x}\left(t^{2}+1\right) d t$$
Looking ahead: Integrals of \(\tan x\) and \(\cot x\) a. Use a change of variables to show that $$\int \tan x d x=-\ln |\cos x|+C=\ln |\sec x|+C$$ b. Show that $$\int \cot x d x=\ln |\sin x|+C$$.
Consider the right triangle with vertices \((0,0),(0, b),\) and \((a, 0)\) where \(a>0\) and \(b>0 .\) Show that the average vertical distance from points on the \(x\) -axis to the hypotenuse is \(b / 2,\) for all \(a>0\).
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