§4.3 The definite integral

Introduction

The definition of the definite integral

Some properties of the definite integral

How the definite integral is connected to a function's average value

Consider this https://gvsu.edu/s/a9, in which you will initially see the situation shown in the following figure. The screenshot shows a right Riemann sum with 10 subintervals for the function $f(x) = \sin(2x) - \frac{x^2}{10} + 3$ on the interval $[1,7]$. The value of the sum is $R_{10} = 4.90595$.

Note that the value of the chosen Riemann sum is displayed next to the word “relative,” and that you can change the type of Riemann sum being computed by dragging the point on the slider bar below the phrase “sample point placement.” Explore to see how you can change the window in which the function is viewed, as well as the function itself. You can set the minimum and maximum values of $x$ by clicking and dragging on the blue points that set the endpoints; you can change the function by typing a new formula in the “f(x)” window at the bottom; and you can adjust the overall window by “panning and zooming” by using the Shift key and the scrolling feature of your mouse. More information on how to pan and zoom http://gvsu.edu/s/Fl.

Update the interactive (and view window, as needed) so that the function being considered is $f(x) = 2x+1$ on $[1,4]$. When you do so, you should see the following figure.

For this function on this interval, compute $L_{n}$, $M_{n}$, $R_{n}$ for $n = 5$, $n = 25$, and $n = 100$. What appears to be the exact area bounded by $f(x) = 2x+1$ and the $x$-axis on $[1,4]$?

$\int_0^1 3x \, dx$

$\int_{-1}^4 (2-2x) \, dx$

$\int_{-1}^1 \sqrt{1-x^2} \, dx$

$\int_{-3}^4 g(x) \, dx$, where $g$ is the function pictured in the following figure. Assume that each portion of $g$ is either part of a line or part of a circle.

$\int_5^2 f(x) \, dx$

$\int_0^5 g(x) \, dx$

$\int_0^5 (f(x) + g(x))\, dx$

$\int_2^5 (3x^2 - 4x^3) \, dx$

$\int_5^0 (2x^3 - 7g(x)) \, dx$

Sketch an accurate graph of $y = v(t)$. What kind of curve is $y = \sqrt{4-(t-2)^2}$?

Evaluate $\int_0^4 v(t) \, dt$ exactly.

In terms of the physical problem of the moving object with velocity $v(t)$, what is the meaning of $\int_0^4 v(t) \, dt$? Include units on your answer.

Determine the exact average value of $v(t)$ on $[0,4]$. Include units on your answer.

Sketch a rectangle whose base is the line segment from $t=0$ to $t = 4$ on the $t$-axis such that the rectangle's area is equal to the value of $\int_0^4 v(t) \, dt$. What is the rectangle's exact height?

How can you use the average value you found in (d) to compute the total distance traveled by the moving object over $[0,4]$?