Chapter 1 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   1.
   

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2. Original problem

   2.
   

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3. Original problem

   3.
   

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4. Original problem

   4.
   

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5. Original problem

   5.
   

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6. Original problem

   6.
   

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7. Original problem

   7. Find the arclength of from .
   

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8. Original problem · clarified

   8. Parametrize an ellipse centered at the origin, with a horizontal major axis of 4 units and a minor axis of 2 units, which has a clockwise orientation
   

   Clarification: the ellipse is centered at the origin and its major axis is horizontal. These conventions make the problem agree with the stated parametrization.

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9. Original problem

   9.
   

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10. Original problem

    10.

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11. Added problem

    Evaluate $\int \frac{1}{\sqrt{9-x^2}}dx$ on .

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12. Added problem

    Evaluate $\int \frac{x^2}{\sqrt{4-x^2}}dx$ on .

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13. Added problem

    Evaluate $\int \sqrt{x^2+4}dx$.

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14. Added problem

    Evaluate ${\int }_0^3\frac{1}{(x^2+9{)}^{3/2}}dx$.

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15. Added problem

    Evaluate $\int x\ln xdx$ for $x>0$.

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16. Added problem

    Evaluate $\int x^2\cos xdx$.

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17. Added problem

    Evaluate $\int e^x\sin xdx$.

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18. Added problem

    Evaluate ${\int }_1^e\ln xdx$.

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19. Added problem

    Evaluate $\int \frac{3x+5}{x^2+x-2}dx$ on intervals avoiding the poles.

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20. Added problem

    Evaluate $\int \frac{1}{x(x+1{)}^2}dx$ for $x\ne 0,-1$.

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21. Added problem

    Evaluate $\int \frac{x^3}{x^2+1}dx$.

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22. Added problem

    Evaluate $\int \frac{x+1}{(x-1)(x^2+1)}dx$ for $x\ne 1$.

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23. Added problem

    Eliminate the parameter from $x=2t+1$, $y=t^2$, , and describe the direction of travel.

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24. Added problem

    Find the tangent line to $x=t^2$, $y=t^3+t$ at $t=1$.

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25. Added problem

    Find the arclength of $x=t^2$, $y=\frac{2}{3}{t}^3$ for $0\le t\le 1$.

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26. Added problem

    Parametrize $\frac{(x-1{)}^2}{9}+\frac{(y+2{)}^2}{4}=1$ counterclockwise, beginning at its rightmost point.

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27. Added problem

    Determine convergence and evaluate ${\int }_1^{\infty }\frac{1}{x^3}dx$ if it converges.

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28. Added problem

    Determine convergence and evaluate ${\int }_0^1\frac{1}{\sqrt{x}}dx$ if it converges.

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29. Added problem

    Determine convergence and evaluate ${\int }_0^1\ln xdx$ if it converges.

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30. Added problem

    Does the ordinary improper integral ${\int }_0^2\frac{1}{x-1}dx$ converge? Explain using its one-sided limits.

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