1. [latex]\int_{-2}^2 f(x)dx[/latex].
   
   [latex]2[/latex]

   ans
2. [latex]\int_{2}^7 f(x)dx[/latex].
   
   [latex]8[/latex]

   ans
3. [latex]\int_0^6 f(x)dx[/latex].
   
   [latex]8[/latex]

   ans
4. [latex]\int_4^7 f(x)dx[/latex].
   
   [latex]0[/latex]

   ans
5. [latex]\int_{-2}^7 f(x)dx[/latex].
   
   [latex]10[/latex]

   ans
6. [latex]\int_{3}^3 f(x)dx[/latex] (what do you think this even means?)
   
   [latex]0[/latex] — You can think of this as an infinitely thin rectangle.

   ans

7. [latex]\int_{-1}^{0} f(x)dx[/latex].
   
   [latex]1[/latex]

   ans
8. [latex]\int_{0}^{1.5} f(x)dx[/latex].
   
   [latex]-1/2[/latex]

   ans
9. [latex]\int_{-1.5}^{4.5} f(x)dx[/latex]
   
   [latex]10.5[/latex]

   ans

1. Draw the position graph corresponding to this velocity graph next to the graph above.
   
   

   ans
2. Find the following three integrals based on the original graph [latex]v(t)[/latex]:

   [latex]\int_0^3 v(t)dt, \quad \int_0^6 v(t)dt, \quad \int_0^9 v(t)dt.[/latex]

   [latex]60[/latex], [latex]30[/latex], [latex]30[/latex]

   ans
3. Your answers for parts (a) and (b) should look the same in some way. How do they look the same, and why did it work out this way?
   
   The answers are the key [latex]y[/latex]-values from the graph in part (a). This is the same, since in both cases you are doing the same thing: takeing [latex]\Delta x \cdot y[/latex] (or [latex]\Delta t \cdot \text{velocity}[/latex] and adding it up as you go.

   ans

1. [latex]e(x) = 2 - x[/latex].
   
   [latex]\approx 2.25[/latex]

   ans

2. [latex]f(x) = -\frac{1}{9} x^2 + 1[/latex]
   
   [latex]\approx 2.24[/latex]

   ans
3. [latex]g(x) = e^x[/latex]
   
   [latex]\approx 14.71[/latex]

   ans

1. If you haven’t done so already, sketch a picture of the graph as well as the rectangles you used to approximate the area.
2. Is your approximation from problem 3 an over-estimate or an under-estimate? How do you know?
   
   Overestimate, since it looks like the rectangles cover too much area.

   ans

1. [latex]e(x) = 2 - x[/latex].
   
   [latex]1.5[/latex]

   ans
2. [latex]f(x) = -\frac{1}{9} x^2 + 1[/latex]
   
   [latex]1.99[/latex]

   ans
3. [latex]g(x) = e^x[/latex]
   
   [latex]19.48[/latex]

   ans

1. [latex]e(x) = 2 - x[/latex].
   
   [latex]1.5[/latex]

   ans
2. [latex]f(x) = -\frac{1}{9} x^2 + 1[/latex]
   
   [latex]2.03[/latex]

   ans
3. [latex]g(x) = e^x[/latex]
   
   [latex]18.31[/latex]

   ans

1. [latex]e(x) = 2 - x[/latex].
   
   [latex]1.5[/latex]

   ans
2. [latex]f(x) = -\frac{1}{9} x^2 + 1[/latex]
   
   [latex]2[/latex]

   ans
3. [latex]g(x) = e^x[/latex]
   
   [latex]\approx 19.09[/latex]

   ans

1. [latex]1.5[/latex] for [latex]e(x) = 2-x[/latex].
2. [latex]2[/latex] for [latex]f(x) = -\frac{1}{9} x^2 + 1[/latex]
3. [latex]19.086[/latex] for [latex]g(x) = e^x[/latex].

Given these answers, rate the following rules from most accurate to least accurate based on the answers from this homework: left rectangle rule, trapezoid rule, midpoint rule, Simpson’s rule.