Estimate 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 of the function 25 - x^2 from x = 0 to x = 5 using 5 left rectangles, we can use the Left Riemann Sum.
The Left Riemann Sum approximates the area under a curve by dividing the interval [0, 5] into equal subintervals and approximating each subinterval with the leftmost point. In this case, we have 5 equal subintervals since we want to use 5 left rectangles.
Let's break down the steps to find the estimate:
Step 1: Determine the width of each subinterval.
Since we have 5 subintervals dividing the interval [0, 5], each subinterval will have a width of (5 - 0) / 5 = 1.
Step 2: Calculate the left endpoint of each subinterval.
Starting from the left endpoint 0, we can calculate the left endpoint of each subinterval by adding the width successively. The left endpoints for the 5 subintervals are:
0, 1, 2, 3, and 4.
Step 3: Evaluate the function at each left endpoint.
Evaluate the function 25 - x^2 at each left endpoint of the subintervals:
For the leftmost point (0): f(0) = 25 - 0^2 = 25
For the second leftmost point (1): f(1) = 25 - 1^2 = 24
For the third leftmost point (2): f(2) = 25 - 2^2 = 21
For the fourth leftmost point (3): f(3) = 25 - 3^2 = 16
For the fifth leftmost point (4): f(4) = 25 - 4^2 = 9
Step 4: Calculate the sum of the areas of the rectangles.
The area of each rectangle is given by the width multiplied by the height. In this case, the height is the value of the function at each left endpoint.
Summing up the areas of the rectangles, we get:
(1)(25) + (1)(24) + (1)(21) + (1)(16) + (1)(9) = 25 + 24 + 21 + 16 + 9 = 95.
The estimated area under the graph of the function 25 - x^2 from x = 0 to x = 5 using 5 left rectangles is 95 square units.