91Ó°ÊÓ

Evaluating a Sum In Exercises \(25-28,\) use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for \(n=10,100,1000,\) and \(10,000 .\) \(\sum_{i=1}^{n} \frac{2 i+1}{n^{2}}\)

Approximating the Area of a Plane Region In Exercises \(29-34,\) use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the \(x\) -axis over the given interval. \(g(x)=\sin x,[0, \pi], 6\) rectangles

In Exercises 41- 46, find the area of the region bounded by the graphs of the equations. $$y=5 x^{2}+2, \quad x=0, \quad x=2, \quad y=0$$

Finding Upper and Lower Sums for a Region In Exercises \(41-44,\) find the upper and lower sums for the region bouded by the graph of the function and the \(x\) -axis on the given interval. Leave your answer in terms of \(n,\) the number of subintervals. \(f(x)=5 x^{2} \quad \quad \quad[0,1]\)

Finding Area by the Limit Definition In Exercises \(47-56\) , use the limit process to find the area of the region bounded by the graph of the e function and the \(x\) -axis over the given interval. Sketch the region. \(y=x^{2}+2,[0,1]\)

Tree Growth An evergreen nursery usually sells a certain type of shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by \(d h / d t=1.5 t+5,\) where \(t\) is the time in years and \(h\) is the height in centimeters. The seedlings are 12 centimeters tall when planted \((t=0)\) . (a) Find the height after \(t\) years. (b) How tall are the shrubs when they are sold?

Change of Variables In Exercises 53-60, find the indefinite integral by making a change of variables. $$\int \frac{x^{2}-1}{\sqrt{2 x-1}} d x$$

With what initial velocity must an object be thrown upward (from a height of 2 meters) to reach a maximum height of 200 meters?

Buffon's Needle Experiment A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is $$P=\frac{2}{\pi} \int_{0}^{\pi / 2} \sin \theta d \theta$$ where \(\theta\) is the acute angle between the needle and any one of the parallel lines. Find this probability.

Lunar Gravity On the moon, the acceleration of a free-falling object is \(a(t)=-1.6\) meters per second per second. A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?