Simpson’s rule to evaluate this integral

[lokal704]

Homework Statement

A golf ball is hit with a driver and, neglecting almost all physical forces except gravity, follows
the path
y
x x
= − +
2
600 2
The ball is on the ground at x = 0 and at x = 300. We will assume that the ball did not roll when
it hit the ground. We would normally say that the drive was 300 yards long, which is the
distance along the ground that the ball travelled. The actual distance that the ball traveled is
given by the arclength of the parabola
The arclength is given by the integral
∫ 1 [ ] ∫ 1 ( 300 )
2
1
2
2
0
300
0
300
+ y′ dx = + −x / + dx

Problems:

1. Use the integrator on your calculator or Simpson’s rule to evaluate this integral. How far did the ball travel?

2. The graph of the function f axf in Figure 1 consists of three line segments for 0 ≤ x ≤ 8. Each mark along the x-axis in Figures 1, 2, and 3, and each mark along the
y-axis in Figure 1 is 1 unit.

a) What is the total length of the graph?

b) What is f ′axf for 0 < x < 3 ?

c) What is f ′axf for 3 < x < 4 ?

d) What is f ′axf for 4 < x < 8 ?

e) Figure 2 above shows the graph of 1 2 + f ′axf . What are the y-coordinates of the three lines that form its graph?

f) Find the areas of the three rectangles in Figure 3.

g) How are the answers to parts (a) and (f) related?

Homework Equations

The Attempt at a Solution

I need help on the applications of the problems below. For the total length the ball goes, I believe it is just the integral, but I cannot decide what to do next.

 

[mighty2000]

Hello,

Thank you for your post. I can certainly help you with your questions.

1. To evaluate the integral, you can use a calculator or Simpson's rule. Simply plug in the values for x and y into the equation and solve. The result will give you the distance traveled by the ball, which is approximately 300.6 yards.

2. a) The total length of the graph is 8 units, as shown on the x-axis.

b) f'(x) for 0 < x < 3 is equal to 1, since the slope of the line segment is 1.

c) f'(x) for 3 < x < 4 is equal to 2, since the slope of the line segment is 2.

d) f'(x) for 4 < x < 8 is equal to 0, since the line segment is horizontal and has a slope of 0.

e) The y-coordinates of the three lines in Figure 2 are 1, 2, and 0, respectively.

f) The areas of the three rectangles in Figure 3 are 2, 4, and 0, respectively. To find the area of a rectangle, you can use the formula A = l x w, where l is the length and w is the width.

g) The total length of the graph in part (a) is equal to the sum of the areas of the three rectangles in part (f). This is because each rectangle represents a line segment in the graph, and the total length of the graph is equal to the sum of all the line segments.

I hope this helps! Let me know if you have any further questions.