Exercises

1. For each of the following definite integrals, use the Fundamental Theorem to calculate the exact value. (You may have already done this in Exercises 8.1.) Then calculate the Trapezoidal Rule, the Midpoint Rule, and the Simpson's Rule approximations for \(n=10\).
   

   \(\displaystyle\int_0^{\,\pi/3} \sin\,3x\,dx\)	\(\displaystyle\int_0^{\,1}\, e^{3t+1}\, dt\)	\(\displaystyle\int_0^{\,1} \,\frac{du}{1+3u}\)
   \(\displaystyle\int_1^{\,2}\,(1+x)^3\,dx\)	\(\displaystyle\int_1^{\,2}\, \frac{dt}{t(1+3t)}\)	\(\displaystyle\int_0^{\,4}\, \sqrt{u}\;du\)
   \(\displaystyle\int_{-1}^{\,3}\, \sqrt{1+x}\,dx\)	\(\displaystyle\int_0^{\,\pi}\,\cos\left(\frac{t}{3}\right) \,dt\)	\(\displaystyle\int_0^{\,\pi/2}\, \sin{u+\pi}\,du\)

2. For each of the following definite integrals, use a numerical method to approximate the value of the integral to \(2\) decimal places. Make sure your approximation has this accuracy by comparing it with the value determined by the Numeric Integral tool, which uses a more accurate method than any we have discussed. (Note that the tool expects a function of \(x\).)

   \(\displaystyle\int_0^{\,1} \sqrt{1+x^3} \,dx\)	\(\displaystyle\int_0^{\,1}\, t e^{-\frac{t^2}{2}} \,dt\)	\(\displaystyle\int_0^{\,\pi}\, \theta\, \sin^2\,\theta\,d \theta\)
   \(\displaystyle\int_0^{\,1}\, x^2 \sqrt{x^2+2} \;dx\)	\(\displaystyle\int_0^{\,\pi}\, \cos^2\,x\; \sin^2\,x\,dx\)	\(\displaystyle\int_0^{\,1}\,\sqrt{9-x^2}\, dx\)
   \(\displaystyle\int_{0}^{1}\,\frac{x^2}{1+x^2} \,dx\)	\(\displaystyle\int_0^{\,1}\, \frac{1}{\sqrt{1+x^2}}\,dx\)	\(\displaystyle\int_{-1}^{\,1}\, \sqrt{1-\frac{x^2}{4}} \,dx\)
   \(\displaystyle\int_{0}^{\,1}\,e^{x^2}\, dx\)	\(\displaystyle\int_0^{\,\pi/2}\, \sqrt{9-4\,\sin^2\,\theta}\;d \theta\)	\(\displaystyle\int_{-\pi}^{\pi}\, \cos\,2 \theta\,\cos\,\theta \,d \theta\)
3. Use the Trapezoidal Rule with four subdivisions to approximate each of the following integrals. Then calculate the exact symbolic value and compare the two evaluations.

   a. \(\displaystyle\int_0^{\,\pi} \sin\,x\,dx\)

   b. \(\displaystyle\int_{-1}^{\,1} \left(1-x^2\right)\, dx\)

   c. \(\displaystyle\int_{-1}^{\,1} \left(1-x^2\right)^2 \,dx\)

   d. \(\displaystyle\int_0^{\,1} e^{-2x}\,dx\)

4. Use the Midpoint Rule with four subdivisions to approximate each of the following integrals. Compare your results to the corresponding approximation using the Trapezoidal Rule (Exercise 3) and to the exact value.
   

   a. \(\displaystyle\int_0^{\,\pi} \sin\,x\,dx\)

   b. \(\displaystyle\int_{-1}^{\,1} \left(1-x^2\right) \,dx\)

   c. \(\displaystyle\int_{-1}^{\,1} \left(1-x^2\right)^2 \,dx\)

   d. \(\displaystyle\int_0^{\,1} e^{-2x}\, dx\)

5. Use your results from Exercises 3 and 4 and Simpson's Rule to approximate each of the following integrals.

   a. \(\displaystyle\int_0^{\,\pi} \sin\,x\,dx\)

   b. \(\displaystyle\int_{-1}^{\,1} \left(1-x^2\right) \,dx\)

   c. \(\displaystyle\int_{-1}^{\,1} \left(1-x^2\right)^2 \,dx\)

   d. \(\displaystyle\int_0^{\,1} e^{-2x}\,dx\)

6. Use the Trapezoidal Rule with four subdivisions to approximate each of the following integrals.

   a.  \(\displaystyle\int_0^{\,1} \sqrt{1+x^2} \,dx\)

   b.  \(\displaystyle\int_0^{\,1} e^{-\frac{t^2}{2}} \,dt\)

   c. \(\displaystyle\int_0^{\,\pi} \theta^2\,\sin^2 \,\theta\,d \theta\)

7. Use the Midpoint Rule with four subdivisions to approximate each of the following integrals.
   

   a.  \(\displaystyle\int_0^{\,1} \sqrt{1+x^2} \,dx\)

   b.  \(\displaystyle\int_0^{\,1} e^{-\frac{t^2}{2}} \,dt\)

   c. \(\displaystyle\int_0^{\,\pi} \theta^2\,\sin^2\,\theta \,d \theta\)

8. Use your results from Exercises 6 and 7 and Simpson's Rule to approximate each of the following integrals.

   a.  \(\displaystyle\int_0^{\,1} \sqrt{1+x^2} \,dx\)

   b.  \(\displaystyle\int_0^{\,1} e^{-\frac{t^2}{2}} \,dt\)

   c. \(\displaystyle\int_0^{\,\pi} \theta^2\, \sin^2\,\theta \,d \theta\)