Answer:
Explanation:
To approximate the area under the curve of the function y = 20x - x^2 in the interval [0, 20], we can use rectangles with a specific width and height. The area of each rectangle will be the product of its width and height. The sum of the areas of all the rectangles will give us an approximation of the total area under the curve.
Here are the steps to follow:
1. Sketch the graph of the function y = 20x - x^2. You can use a graphing calculator or software to do this.
2. Divide the interval [0, 20] into the given number of rectangles. For example, if you are given 4 rectangles, you can divide the interval into 4 equal parts of width 5.
3. For each rectangle, choose a sample point within the interval and evaluate the function at that point to find its height. You can choose the left endpoint, the right endpoint, or the midpoint of each subinterval as the sample point.
4. Multiply the width and height of each rectangle to find its area.
5. Add up the areas of all the rectangles to get an approximation of the total area under the curve.
For example, let's say we are given 4 rectangles to approximate the area under the curve of y = 20x - x^2 in the interval [0, 20]. We can divide the interval into 4 equal parts of width 5, and choose the left endpoint of each subinterval as the sample point. Then, we can evaluate the function at each sample point to find the height of each rectangle.
The left endpoints of the subintervals are 0, 5, 10, and 15. The heights of the rectangles are:
y(0) = 20(0) - 0^2 = 0
y(5) = 20(5) - 5^2 = 75
y(10) = 20(10) - 10^2 = 100
y(15) = 20(15) - 15^2 = 75
The width of each rectangle is 5, so the areas of the rectangles are:
A1 = 0(5) = 0
A2 = 75(5) = 375
A3 = 100(5) = 500
A4 = 75(5) = 375
The total area under the curve is the sum of the areas of the rectangles:
A = A1 + A2 + A3 + A4 = 0 + 375 + 500 + 375 = 1250
Therefore, the approximation of the area under the curve of y = 20x - x^2
in the interval [0, 20] using 4 rectangles is 1250.