use limit process.

find area of region between graph y= -3x + 20 and the x-axis ove the interval [0,5]

To find the area of the region between the graph of the function y = -3x + 20 and the x-axis over the interval [0, 5], we can use the limit process. The area can be calculated by dividing the interval into small subintervals, finding the area of rectangles under the curve, and taking the limit as the width of the subintervals approaches zero.

Step 1: Divide the interval [0, 5] into n equal subintervals of width Δx = (5 - 0) / n.

Step 2: Choose an arbitrary point within each subinterval, which corresponds to the x-coordinate of the rectangle base.

Step 3: Calculate the height of each rectangle by substituting the chosen point into the function y = -3x + 20.

Step 4: Multiply the width (Δx) and height of each rectangle to find the area of that rectangle.

Step 5: Sum up the areas of all the rectangles.

Step 6: Take the limit of the sum as n approaches infinity (∞). This will give us the exact area under the curve.

Let's calculate the area using a sample value of n = 4:

For n = 4, Δx = (5 - 0) / 4 = 1.25

Now, we need to find the height (y) of each rectangle for the chosen x-coordinate (xi), where i represents the subinterval number: 1, 2, 3, 4.

(x1, y1) = (0.625, -3(0.625) + 20)
(x2, y2) = (1.875, -3(1.875) + 20)
(x3, y3) = (3.125, -3(3.125) + 20)
(x4, y4) = (4.375, -3(4.375) + 20)

Calculate the area of each rectangle:
A1 = (1.25)(-3(0.625) + 20)
A2 = (1.25)(-3(1.875) + 20)
A3 = (1.25)(-3(3.125) + 20)
A4 = (1.25)(-3(4.375) + 20)

Sum up the areas:
A = A1 + A2 + A3 + A4

Finally, we take the limit as n approaches infinity (∞) to get the exact area between the curve and the x-axis over the interval [0, 5].