7.2: Homework- Power, exponential, trig, and logarithmic rules

Last updated
Save as PDF

Page ID: 88685

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Compute the following definite integrals.
1. \(\int_{2}^3 x^3 + 2 \sqrt{x} dx\)
  19.04
  ans
2. \(\int_{-2}^3 (x + 5)^2dx\)
  \(\approx 161.7\)
  ans
3. \(\int_{0}^{1} e^xdx\)
  \(e - 1 \approx 1.718\)
  ans
4. \(\int_{-1}^1 3 e^xdx\)
  \(7.05\)
  ans
5. \(\int_{1}^{e} \frac{3}{x} + \frac{x}{3}\ dx\)
  \(\approx 4.06\)
  ans
Approximate \(\int_0^1 x^2dx\) using \(4\) rectangles. Then find \(\int_0^1 x^2\) exactly using an anti-derivative. How far off is the approximation?
Approximation \(\approx 0.22\), the actual is \(\frac{1}{3} \approx .33\), so the difference is about \(0.11\) or \(50\) error which isn't great. As we know, the rectangles don't always do such a good job.
ans