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Section Solutions

Riemann Sums

  1. Lower = 12.875, Upper = 17.375, Average = 15.125

  2. 15

  3. Upper \(\approx\) -1.836, Lower \(\approx\) -1.903, Average \(\approx\) -1.869

Definite and Indefinite Integrals

  1. Compute the following indefinite integrals.

    1. \(\dsp{x^2 + \frac{1}{4}x^3 + c}\)

    2. \(\dsp s + e^s + c\)

    3. \(\dsp -\cos(x)+c\)

    4. \(\dsp \sin(y)+c\)

    5. \(\dsp \pi \sin(x) + \pi x +c\)

    6. \(\dsp -\frac{1}{2}\cos(2t)+c\)

    7. \(\dsp \frac{10}{3}x^{3/2} - \ln(x) +c\)

    8. \(\dsp \frac{1}{3}x^3+x+c\)

    9. \(\dsp \frac{1}{5}x^5 + \frac{2}{3}x^3+x+c\)

    10. \(\dsp \frac{1}{6}(x^2+1)^3 + c\)

    11. \(\dsp \sin(x^2)+c\)

    12. \(\dsp 5\ln(t) + \frac{1}{5}\ln(t) + \frac{1}{\ln(5)} 5^t+c\)

    13. \(\dsp \frac{1}{5} \sin^5(t) +c\)

    14. \(\dsp \frac{1}{9} (5x^3-3x^2)^9 +c\)

    15. \(\dsp \frac{5}{9}y^9 - \frac{5}{2}y^8 + \frac{20}{7}y^7 - \frac{2}{5}y^5 + 2y^4 - \frac{8}{3}y^3 +c\)

    16. \(\dsp \frac{1}{6}t^6-\frac{3}{4}t^4+c\)

    17. \(\dsp -\frac{\pi}{2}x^2+ex+c\)

  2. Compute the following definite integrals.

    1. \(8\)

    2. \(6\)

    3. \(84\)

  3. Evaluate each integral.

    1. \(\dsp x+{x}^{2} + k\)

    2. \(\dsp \frac{1}{12}{t}^{4}-\frac{2}{3}{t}^{3/2} + k\)

    3. \(\dsp 4 ( \tan(x) - \sin(x) ) +k\)

    4. \(\dsp \frac{2}{5}\sqrt{z^5}-4\,{z}^{-1}+\frac{3}{16} \frac{3}{\sqrt[3]{4}\sqrt[3]{z^2}} + k\)

    5. \(\dsp \frac{1}{3}\sqrt {x} \left( 3+x \right) \sqrt {2} + k\)

    6. \(\dsp -\cos(t) + 2\sqrt(3t) + k\)

    7. \(\dsp -\frac{1}{3}\cos \left( {x}^{3} \right) + k\)

    8. \(\dsp -\frac{1}{6} \left( \cos \left( x \right) \right) ^{6} + k\)

    9. \(\dsp -\frac{1}{3} \left( 1-{x}^{2} \right) ^{3/2} + k\)

    10. \(\dsp \frac{1}{4} \left( 3\,{z}^{2}+6\,z+5 \right) ^{2/3} + k\)

    11. \(\dsp \frac{1}{2}\sin \left( 2 \right) -\frac{1}{2} + k\)

    12. \(1-\sqrt{2} + k\)

    13. \(\dsp {\frac {2145}{4}} + k\)

    14. \(\sqrt{2}-1 + k\)

    15. \(\dsp {\frac {20}{3}}\,\sqrt {5}-{\frac {32}{3}} + k\)

  4. Evaluate each of the following integrals, utilizing the suggested substitution.

    1. \(\dsp 2/3\,\sqrt {x+5} \left( -10+x \right) + k\)

    2. \(\dsp {\frac {3}{28}}\, \left( x-1 \right) ^{4/3} \left( 3+4\,x \right) + k\)

    3. \(\dsp -1/3 \sqrt{4-x^2} (x^2+8) + k\)

    4. \(\dsp 1/3\,\sqrt {{x}^{2}+1} \left( -2+{x}^{2} \right) + k\)

    5. \(\int 1/2\,x\sqrt {9-{x}^{2}}+9/2\,\arcsin \left( 1/3\,x \right) + k\)

    6. \(\dsp 1/4\,\ln \left( x+2 \right) -1/4\,\ln \left( x-2 \right) + k\)

    7. \(\arctan \left( x \right) + k\)

Average Value, Mean Value, Fundamental Theorem and Arclength

  1. \(-5\)

  2. \(\dsp \frac{1}{2}\)

  3. \(\dsp \frac{\sqrt{93}-3}{3} \approx 2.215\)

  4. \(F'(x) =\cos(x)\)

  5. \(F'(x) = x^3 - 4x + e^x \; dt\)

  6. \(\dsp F'(x) = \frac{ e^x}{2\sqrt{x}}\)

  7. \(\dsp \frac{ 74 \sqrt{37}-2}{27} \approx 16.6\)

Area and Volume

  1. \(\dsp \frac{7}{3}\)

  2. \(10\)

  3. \(\dsp \frac{32}{3}\)

  4. \(9\)

  5. \(8\)

  6. \(\dsp \frac{32}{3}\)

  7. \(no\)

  8. \(\dsp \frac{9}{2}\)

  9. \(\dsp \frac{71}{6}\)

  10. \(8 \pi\)

  11. \(\dsp \frac{8 \pi}{3}\)

  12. \(32 \sqrt{2} \pi\)

  13. \(\dsp \frac{32\sqrt{3}}{3}\)

  14. \(\dsp \frac{4\pi r^3}{3}\)

  15. no answer — attempt to estimate whether your answer makes sense

  16. no answer — attempt to estimate whether your answer makes sense

  17. no answer — attempt to estimate whether your answer makes sense

  18. no answer — attempt to estimate whether your answer makes sense

Work

  1. 15/4 ft lbs

  2. 650,000 ft lbs

  3. 2450 J

Center of Mass

  1. \(95\) and approximately \(6.3\)

  2. \(\dsp ( \frac{16}{105}, \frac{4}{15} )\)

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