Integral Calculus by finding area
This slide shows how to take a (t,v) graph and convert it back into a (t,x) graph. First, one finds the area under the curve in a given time segment. For example, on this graph, there is a change in the curve at 10 seconds, 20 seconds, 30 seconds, and 40 seconds. The time limits are therefore 0-10, 10-20, 20-30, etc. The upper and lower bounds in each segment are the curve and the t-axis. Thus, the areas in question are the boxes on the center graph. It is important to recognize that these represent displacement, deltaX, and not position, X. Math can't help you find position from displacement if you don't know your starting position. Look at the two graphs on the right. The (t,v) graph shows a constant velocity of 3 m/s. Which of the (t,x) graphs generated it? The red, the blue, or the green? We don't know. All have a slope of 3 m/s. If I dropped you off in the middle of nowhere on the side of a highway and had you walk 5 miles north, you wouldn't know where you are. You'd be 5 miles north of somewhere you don't know where it is. So someone has to TELL you your starting position, x0. In the table, I specified it as 17 meters. I did that randomly. Notice from the second row of the table downward, I filled in the area of each segment of the curve, delta x. The new position in each row, x, is obtained by adding deltax to whatever x came before it. Hence 50+17=67, 67-30=37, etc. If you think of the table as a checkbook, deltax is a deposit or withdrawal, and x is the balance.
Integral Calculus by finding area
This slide shows how to take a (t,v) graph and convert it back into a (t,x) graph. First, one finds the area under the curve in a given time segment. For example, on this graph, there is a change in the curve at 10 seconds, 20 seconds, 30 seconds, and 40 seconds. The time limits are therefore 0-10, 10-20, 20-30, etc. The upper and lower bounds in each segment are the curve and the t-axis. Thus, the areas in question are the boxes on the center graph. It is important to recognize that these represent displacement, deltaX, and not position, X. Math can't help you find position from displacement if you don't know your starting position. Look at the two graphs on the right. The (t,v) graph shows a constant velocity of 3 m/s. Which of the (t,x) graphs generated it? The red, the blue, or the green? We don't know. All have a slope of 3 m/s. If I dropped you off in the middle of nowhere on the side of a highway and had you walk 5 miles north, you wouldn't know where you are. You'd be 5 miles north of somewhere you don't know where it is. So someone has to TELL you your starting position, x0. In the table, I specified it as 17 meters. I did that randomly. Notice from the second row of the table downward, I filled in the area of each segment of the curve, delta x. The new position in each row, x, is obtained by adding deltax to whatever x came before it. Hence 50+17=67, 67-30=37, etc. If you think of the table as a checkbook, deltax is a deposit or withdrawal, and x is the balance.