An area is bounded by the x-axis and the parabola y = 16 - x^2. Use four rectangles of equal width and the midpoint approximation method to estimate the bounded area.

Could you please show me how to work out this problem?

Thanks!

Sure! To estimate the area bounded by the x-axis and the parabola y = 16 - x^2 using four rectangles of equal width and the midpoint approximation method, you can follow these steps:

1. Determine the width of each rectangle: Since you want to use four rectangles, divide the interval [a, b] into four equal parts. In this case, the interval is bounded by the x-axis, so you need to find the values of x that correspond to the points where the parabola intersects the x-axis.

To find these points, set y = 0 and solve for x:
0 = 16 - x^2
x^2 = 16
x = ± √16
x = ± 4

So the interval [a, b] is [-4, 4]. To find the width of each rectangle, divide the length of the interval by the number of rectangles:
Width = (b - a) / Number of rectangles
Width = (4 - (-4)) / 4
Width = 8 / 4
Width = 2

Therefore, the width of each rectangle is 2 units.

2. Find the height of each rectangle: To use the midpoint approximation method, you need to evaluate the function y = 16 - x^2 at the midpoints of each rectangle.

The midpoints of the rectangles can be found by adding half the width to the left endpoint of each rectangle. Assuming the left endpoint of the first rectangle is a = -4, the midpoints of the four rectangles are:
-4 + 1 = -3
-3 + 1 = -2
-2 + 1 = -1
-1 + 1 = 0

Now, evaluate the function y = 16 - x^2 at each midpoint:
y1 = 16 - (-3)^2 = 7
y2 = 16 - (-2)^2 = 12
y3 = 16 - (-1)^2 = 15
y4 = 16 - (0)^2 = 16

So the heights of the four rectangles are: 7, 12, 15, and 16 units.

3. Calculate the area of each rectangle: Multiply the width and height of each rectangle to find its area.

Area of rectangle 1 = Width * Height = 2 * 7 = 14 square units
Area of rectangle 2 = Width * Height = 2 * 12 = 24 square units
Area of rectangle 3 = Width * Height = 2 * 15 = 30 square units
Area of rectangle 4 = Width * Height = 2 * 16 = 32 square units

4. Sum up the areas of all the rectangles: Add the areas of all four rectangles together to get the estimate for the bounded area.

Estimated bounded area = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4
= 14 + 24 + 30 + 32
= 100 square units

So, the estimated bounded area using four rectangles of equal width and the midpoint approximation method is 100 square units.

Sure! To estimate the bounded area using the midpoint approximation method, we will divide the interval into equal widths and calculate the height of each rectangle using the midpoint of each subinterval.

Step 1: Determine the interval
Since the area is bounded by the x-axis, we need to find the x-values where the parabola intersects the x-axis. To do this, set y = 0 and solve for x:

0 = 16 - x^2
x^2 = 16
x = ±√16
x = ±4

So, the interval for the x-axis is [-4, 4].

Step 2: Determine the width of each rectangle
To divide the interval into four equal widths, we need to find the width of each rectangle. We can do this by dividing the total width of the interval by the number of rectangles:

Width = (Upper bound of interval - Lower bound of interval) / Number of rectangles
Width = (4 - (-4)) / 4
Width = 8 / 4
Width = 2

So, the width of each rectangle is 2.

Step 3: Determine the height of each rectangle
To estimate the height of each rectangle, we will use the midpoint of each subinterval. We need to find the x-coordinate of each midpoint and plug it into the equation of the parabola y = 16 - x^2 to get the corresponding y-coordinate:

Subinterval 1: x = -4 + (0.5 * Width) = -4 + (0.5 * 2) = -4 + 1 = -3
Subinterval 2: x = -4 + (1.5 * Width) = -4 + (1.5 * 2) = -4 + 3 = -1
Subinterval 3: x = -4 + (2.5 * Width) = -4 + (2.5 * 2) = -4 + 5 = 1
Subinterval 4: x = -4 + (3.5 * Width) = -4 + (3.5 * 2) = -4 + 7 = 3

Now, plug each x-coordinate into the equation of the parabola to get the y-coordinate:

For x = -3: y = 16 - (-3)^2 = 16 - 9 = 7
For x = -1: y = 16 - (-1)^2 = 16 - 1 = 15
For x = 1: y = 16 - 1^2 = 16 - 1 = 15
For x = 3: y = 16 - 3^2 = 16 - 9 = 7

So, the heights of the rectangles are 7, 15, 15, and 7.

Step 4: Calculate the area of each rectangle
To calculate the area of each rectangle, multiply the width by the height:

Rectangle 1: Area = Width * Height = 2 * 7 = 14
Rectangle 2: Area = Width * Height = 2 * 15 = 30
Rectangle 3: Area = Width * Height = 2 * 15 = 30
Rectangle 4: Area = Width * Height = 2 * 7 = 14

Step 5: Calculate the estimated bounded area
To estimate the bounded area, sum up the areas of all four rectangles:

Estimated Bounded Area = Sum of Rectangle Areas
Estimated Bounded Area = 14 + 30 + 30 + 14
Estimated Bounded Area = 88

Therefore, the estimated bounded area using four rectangles and the midpoint approximation method is 88 square units.