Math Insight

Integration problems

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  1. Evaluate the following integral: $\displaystyle\int_{1}^{2} - 10 t^{2} + 4 t - 5\, dt=$

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

  2. The volume of a cell is given by the function $v{\left (t \right )}=800 - \frac{40}{3 t + 5}$ $\mu {\rm m}^3$ from $t=1$ to $t=4$ seconds. What is the average volume of the cell during this time frame? If rounding, be sure to use at least $5$ significant digits.

     $\mu {\rm m}^3$

  3. Evaluate the integral: $\displaystyle \int - 4 t^{4} - 3 t^{3} - t\, dt =$

  4. Approximate the definite integral $\displaystyle \int_{1}^{2} t^{3} + \ln{\left (2 t \right )}\, dt$ 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.)

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

  5. Evaluate the following integrals. 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.)
    1. $\displaystyle \int_{1}^{2} e^{- 8 t}\, dt =$
    2. $\displaystyle\int 3 e^{- 5 t}\, dt=$

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


    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.)

  7. Evaluate the following integral: $\displaystyle \int_{3}^{5} - \frac{4}{7 t}\, dt =$

    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.)

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Math 1241, Fall 2025