The document discusses using integrals to calculate distance traveled based on a velocity-time graph. It explains that the area under the graph between two times represents the distance traveled during that interval. Specifically, it asks the reader to: 1) Find the velocity between 30-50 seconds from the given graph and explain why the distance can be found from the rectangular area. 2) Count the squares and find the distance traveled from 0-20 seconds by finding the area under the curved graph. 3) Estimate the definite integral of a function representing a football's cross-sectional area to find its total volume.

1. As you drive on the highway you accelerate to 100 feet per second to pass a truck. After you have passed, you slow down to a more moderate 60 ft/s. The diagram shows the graph of your velocity, v(t) as a function of the number of seconds, t, since you started slowing. Copy the diagram in your notes. What does your velocity seem to be between t = 30 and t = 50 seconds? How far do you travel in the time interval [30,50]? Explain why your answer can be represented as the area of a rectangular region of the graph. Shade this reason.

2. The distance you travel between t = 0 and t = 20 can also be represented as the area of a region bounded by the (curved) graph. Identify and count the number of squares in this region. How many feet of traveling distance does each small square on the graph represent? How far did you go in the time interval [0,20]?

3. Definite Integrals The work you just did involved finding the area underneath the graph of a function. This means that you need to multiply x values with y values that vary with respect to x . This type of product is called a definite integral (of y with respect to x ), and, along with the derivative, is one of the fundamental operations of calculus. We will study many techniques for evaluating definite integrals in this course, but for now our purpose is to talk about what they mean . Thinking of them as the area under the curve is a good start. The function to the right represents the cross-sectional area, y square inches, of a football as a function of the distance, x inches, from one of its ends. Estimate the definite integral of y with respect to x . What are the units of the definite integral? What do you think the definite integral represents?