Use 5 right rectangles to estimate the area of the area under the graph of 25−x 2 from x = 0 to x = 5 using 5 left rectangles?
To estimate the area under the graph using left rectangles, we divide the interval from x = 0 to x = 5 into 5 equal subintervals. Each subinterval will have a width of (5 - 0) / 5 = 1.
To estimate the area of each rectangle, we use the left endpoint of each subinterval to find the height. We plug the left endpoint into the function 25 - x^2 to get the corresponding value of y.
Let's calculate the area under the graph using left rectangles:
1. For the first subinterval [0, 1]:
- Left endpoint: x = 0
- Height: y = 25 - (0)^2 = 25
- Area of the rectangle: width * height = 1 * 25 = 25
2. For the second subinterval [1, 2]:
- Left endpoint: x = 1
- Height: y = 25 - (1)^2 = 24
- Area of the rectangle: width * height = 1 * 24 = 24
3. For the third subinterval [2, 3]:
- Left endpoint: x = 2
- Height: y = 25 - (2)^2 = 21
- Area of the rectangle: width * height = 1 * 21 = 21
4. For the fourth subinterval [3, 4]:
- Left endpoint: x = 3
- Height: y = 25 - (3)^2 = 16
- Area of the rectangle: width * height = 1 * 16 = 16
5. For the fifth subinterval [4, 5]:
- Left endpoint: x = 4
- Height: y = 25 - (4)^2 = 9
- Area of the rectangle: width * height = 1 * 9 = 9
To estimate the total area under the curve using left rectangles, we sum up the areas of all the rectangles:
Total Area ≈ 25 + 24 + 21 + 16 + 9 = 95
Therefore, using 5 left rectangles, the estimated area under the graph of 25 - x^2 from x = 0 to x = 5 is approximately 95.