Chapter 26

Approximate \(\int_{1}^{e} \ln x d x\) using Simpson's rule with \(p=10 . S_{10}=\frac{2 M_{5}+T_{5}}{3} .\) Find an upper bound for the error. At a certain point you'll use the fact that \(e<3\).

Suppose we use right- and left-hand sums to approximate \(\int_{a}^{b} f(t) d t\). We partition the interval \([a, b]\) into \(n\) equal pieces each of length \(\Delta t .\) Let \(R_{n}\) be the right-hand sum using \(n\) subdivisions and \(L_{n}\) be the left-hand sum using \(n\) subdivisions. (a) Show that \(R_{n}=L_{n}+f(b) \Delta t-f(a) \Delta t\). (b) Conclude that \(R_{n}-L_{n}=[f(b)-f(a)] \Delta t=[f(b)-f(a)] \frac{|b-a|}{n}\).

Approximate the following integrals using the trapezoidal rule, the midpoint rule, and Simpson's rule for the specified number of subdivisions and compute error bounds using the formulas given in this section. Compare your answers to the exact answer. (a) \(\int_{0}^{2} x^{2} d x \quad n=4\) (b) \(\int_{0}^{1} \frac{1}{x+1} d x \quad n=4\)

Approximate the following integral using the trapezoidal rule, the midpoint rule, and Simpson's rule for the specified number of subdivisions. Look at \(T_{n}, M_{n}\), and \(S_{2 n}\). Compute error bounds. $$ \int_{1}^{2} \ln x d x \quad \text { (a) } n=4 \text { (b) } n=8 \text { (c) } n=10 $$

Compute \(\int_{1}^{e} \ln x d x\) with error less than \(0.002\). Try, by experimentation, to see how many subdivisions are required for \(M_{n}\) and \(T_{n}\) to differ by less than \(0.002\). How many are required for \(L_{n}\) and \(R_{n}\) to differ by less than \(0.002\) ?

(a) Approximate \(\int_{0}^{1} \cos \left(x^{2}\right) d x\) using \(M_{10}\). Find an upper bound for the error. (b) Approximate \(\int_{0}^{1} \cos \left(x^{2}\right) d x\) using \(S_{20} . S_{20}=\frac{2 M_{10}+T_{10}}{3} .\) Find an upper bound for the error.

Consider \(\int_{1}^{4} \sqrt{x} d x\). (a) Find a value of \(n\) for which \(\left|L_{n}-\int_{a}^{b} f(x) d x\right| \leq 0.01\). (b) Use the value of \(n\) from part (a) to find \(L_{n}\) and \(R_{n}\). (c) Is the average of the left- and right-hand sums larger than the integral, or smaller? (d) Compare your numerical approximations to the answer you get using the Fundamental Theorem of Calculus.

$$ \text { Approximate } \int_{0}^{1} \sqrt{1+x^{4}} d x \text { with error less than } 0.01 \text { . } $$

Give upper and lower bounds for \(\int_{0}^{2} \frac{10}{2+x^{5}} d x\) such that the upper and lower bounds differ by less than \(0.01\).

Give upper and lower bounds for \(\int_{2}^{3} \frac{1}{\ln x} d x\) such that the two bounds differ from one another by less than \(0.05 .\) Explain how you know that the upper bound is indeed an upper bound and the lower bound is indeed a lower bound.

Measurements of the width of a pond are taken every 20 yards along its length. The measurements are: 0 yards, 60 yards, 50 yards, 70 yards, 50 yards, and 30 yards. Approximate the surface area of the pond using the Trapezoidal rule.

Approximate each of the following integrals with error less than \(1 / 100\). (Note: If you look at all the questions and think about a strategy in advance, you will only have to compute two integrals in order to answer all four questions with the desired degree of accuracy. There is no problem with being more accurate than is requested.) Briefly explain what you have done and how many subdivisions you used. (i) \(\int_{0}^{1} e^{-\left(\frac{1}{2}\right) x^{2}} d x\) (ii) \(\int_{1}^{2} e^{-\left(\frac{1}{2}\right) x^{2}} d x\) (iii) \(\int_{0}^{2} e^{-\left(\frac{1}{2}\right) x^{2}} d x\) (iv) \(\int_{-2}^{2} e^{-\left(\frac{1}{2}\right) x^{2}} d x\)

Let \(f(x)=\frac{3}{x^{3}+x}\). Let \(a\) and \(b\) be positive constants, \(0

Approximate \(\int_{-1}^{2} \arctan x d x\) using left- and right-hand sums to obtain an upper and lower bound for the integral with difference less than \(0.05 .\) Save time by graphing \(y=\arctan x\) and using symmetry to simplify the problem.

The function \(f(x)\) is decreasing and concave down on the interval \([3,5]\). Suppose that you use a right-hand sum, \(R_{100}\), a left-hand sum, \(L_{100}\), a trapezoidal sum, \(T_{100}\), and a midpoint sum, \(M_{100}\), all with 100 subdivisions, to estimate \(\int_{3}^{5} f(x) d x .\) Select all of the following that must be true. (a) \(L_{100} \geq R_{100}\) (b) \(\int_{3}^{5} f(x) d x \geq T_{100}\) (c) \(T_{100} \geq R_{100}\) (d) \(T_{100} \geq M_{100}\) (e) \(M_{100} \geq L_{100}\) (f) \(T_{100}=\left(L_{100}+R_{100}\right) / 2\)

Suppose that \(g\) is a differentiable function whose derivative is \(g^{\prime}(x)=\frac{2}{x^{2}+3} .\) Partition \([0,2]\) into \(n\) equal pieces each of length \(\Delta x\) and let \(x_{k}=k \Delta x\), where \(k=0,1, \ldots, n .\) Put the following expressions in ascending order (with "<" or "=" signs between them). \(A=\sum_{i=1}^{n} g\left(x_{i}\right) \Delta x \quad B=\sum_{i=0}^{n-1} g\left(x_{i}\right) \Delta x\) \(C=\lim _{n \rightarrow \infty} \sum_{i=0}^{n-1} g\left(x_{i}\right) \Delta x \quad D=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} g\left(x_{i}\right) \Delta x\)

Suppose \(L_{10}<\int_{a}^{b} f(x) d x

Two of the following three integrals you can evaluate exactly. One you cannot, until learning integration by parts. Identify the one you cannot evaluate exactly and approximate it with an error under \(0.05 .\) Find exact answers for the other two integrals. (a) \(\int_{1}^{e} \frac{\ln x}{x} d x\) (b) \(\int_{0}^{1} x e^{x^{2}} d x\) (c) \(\int_{0}^{1} x e^{x} d x\)