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Chapter 5: Problem 33
The height of an arch above the ground is given by the function \(y=10 \sin x,\) for \(0 \leq x \leq \pi\) What is the average height of the arch above the ground?
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The following functions describe the velocity of a car (in mi/hr) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval \([0, t],\) where \(0 \leq t \leq 3\). $$v(t)=\left\\{\begin{array}{ll} 40 & \text { if } 0 \leq t \leq 1.5 \\ 50 & \text { if } 1.5 < t \leq 3 \end{array}\right.$$
Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=\cos \pi x ; a=0, b=\frac{1}{2}, c=1$$
Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{4} \frac{x-2}{\sqrt{x}} d x$$
Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\sqrt{3}} \frac{3 d x}{9+x^{2}}$$
General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f\). Assume \(f\) and its derivatives are continuous for all real numbers. \(\int\left(f^{(p)}(x)\right)^{n} f^{(p+1)}(x) d x,\) where \(p\) is a positive integer, \(n \neq-1\)
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