1. Using midpoint method, estimate the area under the graph of y= sqrt x between x = 0 to x = 1, using 4 rectangles.

3. A soccer ball is kicked straight up into the air from the ground with an initial velocity of 200 ft/s. The velocity of the ball at any moment is shown with v(t)=200 - 32 ft/s.

How would you find:
a) an upper estimate for the velocity after 5 sec have elapsed.

b) a lower estimate for the height attained after 5 sec.

for y=√x the midpoints with 4 rectangles are 1/8, 3/8, 5/8, 7/8

So, the area is 1/4 (√(1/8) + √(3/8) + √(5/8) + √(7/8)) = (1+√3+√5+√7)/(8√2)

I think you mean
v(t)=200 - 32t
not sure why you'd want an estimate for velocity, since you have the formula. Clearly, 200 is an upper estimate, since v decreases with time.

For distance, using right-side rectangles of width 1, the distance would be greater than
1(v(1)+v(2)+v(3)+v(4)+v(5))

1. To estimate the area under the graph of y = √x using the midpoint method and 4 rectangles, follow these steps:

Step 1: Determine the width of each rectangle. In this case, we are using 4 rectangles, so the width of each rectangle will be (1 - 0)/4 = 1/4.

Step 2: Determine the height of each rectangle. The height of each rectangle is determined by evaluating the function y = √x at the midpoint of each rectangle. Since we have 4 rectangles, the midpoints will be 1/8, 3/8, 5/8, and 7/8 within the interval [0, 1].

The heights of the rectangles are:
Rectangle 1: y(1/8) = √(1/8)
Rectangle 2: y(3/8) = √(3/8)
Rectangle 3: y(5/8) = √(5/8)
Rectangle 4: y(7/8) = √(7/8)

Step 3: Calculate the area of each rectangle. The area of each rectangle is the product of its width and height.

Rectangle 1: (1/4) * √(1/8)
Rectangle 2: (1/4) * √(3/8)
Rectangle 3: (1/4) * √(5/8)
Rectangle 4: (1/4) * √(7/8)

Step 4: Sum up the areas of all the rectangles to get the estimated area under the curve. In this case, you would add the areas of all four rectangles:

Estimated area = (1/4) * √(1/8) + (1/4) * √(3/8) + (1/4) * √(5/8) + (1/4) * √(7/8)

2. For the soccer ball problem:
a) To find an upper estimate for the velocity after 5 seconds, substitute t = 5 into the velocity function v(t):

v(5) = 200 - 32(5)

Simplify the equation:

v(5) = 200 - 160

The upper estimate for the velocity after 5 seconds is v(5) = 40 ft/s.

b) To find a lower estimate for the height attained after 5 seconds, integrate the velocity function v(t) with respect to time from t = 0 to t = 5:

∫(0 to 5) (200 - 32t) dt

Evaluate the integral using the limits:

[200t - 16t^2/2] evaluated from 0 to 5

Simplify further:

[200(5) - 16(5)^2/2] - [200(0) - 16(0)^2/2]

Calculate the expression:

(1000 - 400) - (0 - 0)

The lower estimate for the height attained after 5 seconds is 600 ft.