91Ó°ÊÓ

Use Example 1 as a model to evaluate the limit $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(c_{i}\right) \Delta x_{i}$$ over the region bounded by the graphs of the equations. $$ f(x)=\sqrt{x}, \quad y=0, \quad x=0, \quad x=3 $$ (Hint: Let \(c_{i}=3 i^{2} / n^{2}\).)

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n\). Round your answer to four decimal places and compare the results with the exact value of the definite integral. $$ \int_{0}^{2} x^{3} d x, \quad n=8 $$

Evaluate the definite integral by the limit definition. $$ \int_{-2}^{3} x d x $$

Complete the table. Original Integral $$\int x\left(x^{2}+3\right) d x$$

Approximate the definite integral using the Trapezoidal Rule and Simpson's Rule with \(n=4\). Compare these results with the approximation of the integral using a graphing utility. $$ \int_{0}^{\pi / 4} x \tan x d x $$

Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral \((a>0, r>0)\). $$ \int_{-1}^{1}(1-|x|) d x $$

Find the indefinite integral and check the result by differentiation. $$\int\left(t^{2}-\sin t\right) d t$$

Find the value(s) of \(c\) guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. $$ f(x)=\cos x, \quad[-\pi / 3, \pi / 3] $$

Use the table of values to estimate \(\int_{0}^{6} f(x) d x\) Use three equal subintervals and the (a) left endpoints, (b) right endpoints, and (c) midpoints. If \(f\) is an increasing function, how does each estimate compare with the actual value? Explain your reasoning. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & -6 & 0 & 8 & 18 & 30 & 50 & 80 \\ \hline \end{array} $$

A differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, select the MathGraph button.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). $$ \frac{d y}{d x}=-\frac{1}{x^{2}}, x>0,(1,3) $$