Chapter 5.9 Practice

Questions

  1. Calculate the area of each region \(R\) formed by bounding the following.
    1. \(y=x\), \(x=0\), \(x=6\) and the \(x\)-axis.
    2. \(y=-x+6\), \(x=0\), \(x=6\) and the \(x\)-axis.
    3. \(y=4\), \(x=-2\), \(x=3\) and the \(x\)-axis.
    4. \(y=\sqrt{36-x^2}\), \(x=-6\), \(x=6\) and the \(x\)-axis.
    5. \(y=3\sqrt{36-x^2}\), \(x=-6\), \(x=6\) and the \(x\)-axis.
  2. Approximate the region \(R\) given below by using \(n\) left-endpoint rectangles, \(n\) right-endpoint rectangles, and \(n\) midpoint rectangles.
    1. The region \(R\) formed by bounding \(y=x^3\) and \(x\)-axis on the interval \([0,1]\). Use \(n=6\) rectangles.
    2. The region \(R\) formed by bounding \(y=\frac{1}{x}\) and \(x\)-axis on the interval \([1,3]\). Use \(n=4\) rectangles.
  3. Calculate the area of each region \(R\) given below using the limit definition of area.
    1. The region \(R\) formed by bounding \(y=x^2\), the \(x\)-axis, \(x=1\) and \(x=2.\)
    2. The region \(R\) formed by bounding \(y=x^3\), the \(x\)-axis, \(x=0\) and \(x=1.\)
    3. The region \(R\) formed by bounding \(y=2x^2\), the \(x\)-axis, \(x=1\) and \(x=2.\)
    4. The region \(R\) formed by bounding \(y=-x^2+1\), the \(x\)-axis, \(x=-1\) and \(x=1.\)
  4. Calculate the following by interpreting the definite integral as signed area.
    1. \(\displaystyle\int_{1}^{2}x^2\,\mathrm{d}x\)
    2. \(\displaystyle\int_{0}^{1}x^3\,\mathrm{d}x\)
    3. \(\displaystyle\int_{1}^{2}2x^2\,\mathrm{d}x\)
    4. \(\displaystyle\int_{-1}^{1}(-x^2+1)\,\mathrm{d}x\)
    5. \(\displaystyle\int_{-6}^{6}\sqrt{36-x^2}\,\mathrm{d}x\)
    6. \(\displaystyle\int_{-6}^{6}3\sqrt{36-x^2}\,\mathrm{d}x\)
  5. Suppose that \(f\) and \(g\) are integrable and \[\displaystyle\int^{7}_{-1}f(x)\,\mathrm{d}x=3\quad\text{and}\quad \int^{7}_{-1}g(x)\,\mathrm{d}x=4.\] Calculate the following: \[\int_{-1}^{7}(-f(x)+3g(x)+4x)\,\mathrm{d}x.\]
  6. Suppose that \(f\) is integrable and \[3\leq f(x)\leq 5\] on the interval \([-1,2].\) Calculate an upper and lower bound for the following:
    1. \(\displaystyle\int_{0}^{1}f(x)\,\mathrm{d}x\)
    2. \(\displaystyle\int_{-1}^{2}f(x)\,\mathrm{d}x\)
    3. \(\displaystyle\int_{-1}^{2}5f(x)\,\mathrm{d}x\)
    4. \(\displaystyle\int_{-1}^{2}(f(x)+1)\,\mathrm{d}x\)
    5. \(\displaystyle\int_{-1}^{2}(f(x)+x)\,\mathrm{d}x\)

Answers

    1. \(18\)
    2. \(18\)
    3. \(20\)
    4. \(18\pi\)
    5. \(54\pi\)
    1. Right approximation is \(\frac{49}{144}\), left approximation is \(\frac{25}{144}\), midpoint approximation is \(\frac{71}{288}\)
    2. Right approximation is \(\frac{19}{20}\), left approximation is \(\frac{77}{60}\), midpoint approximation is \(\frac{3776}{3465}\)
    1. \(\frac{7}{3}\)
    2. \(\frac{1}{4}\)
    3. \(\frac{14}{3}\)
    4. \(\frac{4}{3}\)
    1. \(\frac{7}{3}\)
    2. \(\frac{1}{4}\)
    3. \(\frac{14}{3}\)
    4. \(\frac{4}{3}\)
    5. \(18\pi\)
    6. \(54\pi\)
  5. \(105\)
    1. \(3\leq\displaystyle\int_{0}^{1}f(x)\,\mathrm{d}x\leq 5\)
    2. \(9\leq \displaystyle\int_{-1}^{2}f(x)\,\mathrm{d}x\leq 15\)
    3. \(45\leq \displaystyle\int_{-1}^{2}5f(x)\,\mathrm{d}x\leq 75\)
    4. \(12\leq\displaystyle\int_{-1}^{2}(f(x)+1)\,\mathrm{d}x\leq 18\)
    5. \(10.5\leq \displaystyle\int_{-1}^{2}(f(x)+x)\,\mathrm{d}x \leq 16.5\)

Return

© Copyright 2025 by the POC Writing Team: Bryan Carrillo, Yat Sun Poon, and David Weisbart. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the POC Writing Team.