CLP-2 Integral Calculus

2

Consider approximating \(\displaystyle\int_2^{10} f(x) \dee{x}\text{,}\) where \(f(x)\) is the function in the graph below.

1. Draw the rectangles associated with the midpoint rule approximation and \(n=4\text{.}\)
2. Draw the trapezoids associated with the trapezoidal rule approximation and \(n=4\text{.}\)

You don't have to give an approximation.

4

Let \(f(x) = x\sin x+2\cos x\text{.}\) Find a reasonable value \(M\) such that \(|f''(x)| \leq M\) for all \(-3 \leq x \leq 2\text{.}\)

6

Give a function \(f(x)\) such that:

• \(f''(x) \leq 3\) for every \(x\) in \([0,1]\text{,}\) and
• the error using the trapezoidal rule approximating \(\displaystyle\int_0^1 f(x) \dee{x}\) with \(n=2\) intervals is exactly \(\dfrac{1}{16}\text{.}\)

7

Suppose my mother is under 100 years old, and I am under 200 years old. ¹⁵We're going somewhere with this. Who is older?

8

1. True or False: for fixed positive constants \(M\text{,}\) \(n\text{,}\) \(a\text{,}\) and \(b\text{,}\) with \(b \gt a\text{,}\)
   \begin{equation*} \dfrac{M}{24}\dfrac{(b-a)^3}{n^2}\leq \dfrac{M}{12}\dfrac{(b-a)^3}{n^2} \end{equation*}
2. True or False: for a function \(f(x)\) and fixed constants \(n\text{,}\) \(a\text{,}\) and \(b\text{,}\) with \(b \gt a\text{,}\) the \(n\)-interval midpoint approximation of \(\displaystyle\int_a^b f(x) \dee{x}\) is more accurate than the \(n\)-interval trapezoidal approximation.

10

Give a polynomial \(f(x)\) with the property that the Simpson's rule approximation of \(\displaystyle\int_a^b f(x) \dee{x}\) is exact for all \(a\text{,}\) \(b\text{,}\) and \(n\text{.}\)

12 (✳)

Find the midpoint rule approximation to \(\displaystyle\int_0^\pi \sin x\dee{x}\) with \(n = 3\text{.}\)

13 (✳)

The solid \(V\) is 40 cm high and the horizontal cross sections are circular disks. The table below gives the diameters of the cross sections in centimeters at 10 cm intervals. Use the trapezoidal rule to estimate the volume of \(V\text{.}\)

14 (✳)

A \(6\) metre long cedar log has cross sections that are approximately circular. The diameters of the log, measured at one metre intervals, are given below:

Use Simpson's Rule to estimate the volume of the log.

15 (✳)

The circumference of an 8 metre high tree at different heights above the ground is given in the table below. Assume that all horizontal cross--sections of the tree are circular disks.

Use Simpson's rule to approximate the volume of the tree.

18 (✳)

The integral \(\displaystyle\int_{-1}^{1} \sin(x^2) \, \dee{x}\) is estimated using the Midpoint Rule with \(1000\) intervals. Show that the absolute error in this approximation is at most \(2\cdot 10^{-6}\text{.}\)

You may use the fact that when approximating \(\int_a^b f(x) \, \dee{x}\) with the Midpoint Rule using \(n\) points, the absolute value of the error is at most \(M(b-a)^3/24n^2\) when \(\left|f''(x)\right|\leq M\) for all \(x\in[a,b]\text{.}\)

20 (✳)

Both parts of this question concern the integral \(I = \displaystyle\int_{0}^{2} (x-3)^5\,\dee{x}\text{.}\)

1. Write down the Simpson's Rule approximation to \(I\) with \(n=6\text{.}\) Leave your answer in calculator-ready form.
2. Which method of approximating \(I\) results in a smaller error bound: the Midpoint Rule with \(n=100\) intervals, or Simpson's Rule with \(n=10\) intervals? You may use the formulas
   \begin{gather*} |E_M| \le \frac{M(b-a)^3}{24n^2} \qquad\text{and}\qquad |E_S| \le \frac{L(b-a)^5}{180n^4}, \end{gather*}
   where \(M\) is an upper bound for \(|f''(x)|\) and \(L\) is an upper bound for \(|f^{(4)}(x)|\text{,}\) and \(E_M\) and \(E_S\) are the absolute errors arising from the midpoint rule and Simpson's rule, respectively.

21 (✳)

Find a bound for the error in approximating \(\displaystyle\int_1^5 \frac{1}{x}\,\dee{x}\) using Simpson's rule with \(n = 4\text{.}\) Do not write down the Simpson's rule approximation \(S_4\text{.}\)

In general the error in approximating \(\int_a^b f(x)\dee{x}\) using Simpson's rule with \(n\) steps is bounded by \(\dfrac{L(b-a)}{180}(\De x)^4\) where \(\De x=\dfrac{b-a}{n}\) and \(L\ge |f^{(4)}(x)|\) for all \(a\le x\le b\text{.}\)

23 (✳)

Let \(I=\displaystyle\int_1^2 (1/x)\,\dee{x}\text{.}\)

1. Write down the trapezoidal approximation \(T_4\) for \(I\text{.}\) You do not need to simplify your answer.
2. Write down the Simpson's approximation \(S_4\) for \(I\text{.}\) You do not need to simplify your answer.
3. Without computing \(I\text{,}\) find an upper bound for \(|I - S_4|\text{.}\) You may use the fact that if \(\big|f^{(4)}(x)\big|\le L\) on the interval \([a, b]\text{,}\) then the error in using \(S_n\) to approximate \(\int_a^b f(x)\,\dee{x}\) has absolute value less than or equal to \(L(b-a)^5/180n^4\text{.}\)

24 (✳)

A function \(s(x)\) satisfies \(s(0)=1.00664\text{,}\) \(s(2)=1.00543\text{,}\) \(s(4)=1.00435\text{,}\) \(s(6)=1.00331\text{,}\) \(s(8)=1.00233\text{.}\) Also, it is known to satisfy \(\big|s^{(k)}(x)\big|\le \dfrac{k}{1000}\) for \(0\le x\le 8\) and all positive integers \(k\text{.}\)

1. Find the best Trapezoidal Rule and Simpson's Rule approximations that you can for \(\displaystyle I=\int_0^8 s(x)\dee{x}\text{.}\)
2. Determine the maximum possible sizes of errors in the approximations you gave in part (a). Recall that if a function \(f(x)\) satisfies \(\big|f^{(k)}(x)\big|\le K_k\) on \([a,b]\text{,}\) then
   \begin{equation*} \bigg|\int_a^b f(x)\dee{x} -T_n\bigg|\le \frac{K_2(b-a)^3}{12n^2} \quad\hbox{and}\quad \bigg|\int_a^b f(x)\dee{x} -S_n\bigg|\le \frac{K_4(b-a)^5}{180n^4} \end{equation*}

26 (✳)

A swimming pool has the shape shown in the figure below. The vertical cross--sections of the pool are semi--circular disks. The distances in feet across the pool are given in the figure at 2--foot intervals along the sixteen--foot length of the pool. Use Simpson's Rule to estimate the volume of the pool.

29 (✳)

Let \(I={\displaystyle\int_0^2}\cos(x^2)\dee{x}\) and let \(S_n\) be the Simpson's rule approximation to \(I\) using \(n\) subintervals.

1. Estimate the maximum absolute error in using \(S_8\) to approximate \(I\text{.}\)
2. How large should \(n\) be in order to ensure that \(|I-S_n|\le 0.0001\text{?}\)

Note: The graph of \(f''''(x)\text{,}\) where \(f(x)=\cos(x^2)\text{,}\) is shown below. The absolute error in the Simpson's rule approximation is bounded by \(\dfrac{L(b-a)^5}{180n^4}\) when \(|f''''(x)|\le L\) on the interval \([a,b]\text{.}\)

30 (✳)

Define a function \(f(x)\) and an integral \(I\) by

Estimate how many subdivisions are needed to calculate \(I\) to five decimal places of accuracy using the trapezoidal rule.

Note that if \(E_n\) is the error using \(n\) subintervals, then \(|E_n|\le\dfrac{M(b-a)^3}{12n^2\vphantom{\frac{1}{2}}}\text{,}\) where \(M\) is the maximum absolute value of the second derivative of the function being integrated and \(a\) and \(b\) are the limits of integration.

31

Let \(f(x)\) be a function ¹⁸For example, \(f(x)=\frac{1}{6}x^3-\frac{1}{2}x^2+(1+x)\log|x+1|\) will do, but you don't need to know what \(f(x)\) is for this problem. with \(f''(x) = \dfrac{x^2}{x+1}\text{.}\)

1. Show that \(|f''(x)| \leq 1\) whenever \(x\) is in the interval \([0,1]\text{.}\)
2. Find the maximum value of \(|f''(x)|\) over the interval \([0,1]\text{.}\)
3. Assuming \(M=1\text{,}\) how many intervals should you use to approximate \(\displaystyle\int_{0}^{1}f(x) \dee{x}\) to within \(10^{-5}\text{?}\)
4. Using the value of \(M\) you found in (b), how many intervals should you use to approximate \(\displaystyle\int_0^1 f(x) \dee{x}\) to within \(10^{-5}\text{?}\)

33

Using an approximation of the area under the curve \(\dfrac{1}{x^2+1}\text{,}\) show that the constant \(\arctan2\) is in the interval \(\left[\dfrac{\pi}{4}+0.321,\, \dfrac{\pi}{4}+0.323\right]\text{.}\)

You may assume use without proof that \(\displaystyle\ddiff{4}{}{x}\left\{\frac{1}{1+x^2}\right\} = \dfrac{24(5x^4-10x^2+1)}{(x^2+1)^5}\text{.}\) You may use a calculator, but only to add, subtract, multiply, and divide.