Math Insight

Integration problems

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  1. The volume of a cell is given by the function $v{\left (t \right )}=600 - 300 e^{- 3 t}$ $\mu {\rm m}^3$ from $t=0$ to $t=2$ seconds. What is the average volume of the cell during this time frame? If rounding, be sure to include at least $5$ significant digits.

     $\mu {\rm m}^3$

  2. Approximate the definite integral $\displaystyle \int_{1}^{4} 3 t^{2} + \ln{\left (3 t \right )}\, dt$ using a right Riemann sum with $3$ intervals. If rounding, be sure to include at least $5$ significant digits. (To be safe, include even more digits.)

    $\displaystyle \int_{1}^{4} 3 t^{2} + \ln{\left (3 t \right )}\, dt=$

  3. Evaluate the following integral: $\displaystyle \int - \frac{3}{7 t + 5}\, dt=$

  4. Evaluate the following integral: $\displaystyle\int_{0}^{1} - 7 t^{2} + 4 t - 8\, dt=$

    If rounding, include at least 5 significant digits in your response.

  5. Find the area bounded by the $x$-axis and the graph of $p{\left (x \right )}=- 3 x^{2} + 5 x + 21$ between $x=0$ and $x=1$.


    For full credit, do not round your answer to a decimal. (Maximum of 80% credit will be awarded for rounded answers that include at least 5 correct significant digits.)

  6. Evaluate the following integrals.
    1. $\displaystyle \int 10 t^{4} + t^{3} + 4 t\, dt =$
    2. $\displaystyle\int - 5 e^{- 9 t}\, dt=$
      Need help?

  7. Evaluate the following integrals.
    1. $\displaystyle \int 6 t^{4} - 7 t^{3} - t\, dt =$
    2. $\displaystyle\int_{-1}^{2} 6 t^{2} + 5 t - 6\, dt=$

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Math 1241, Fall 2020