Estimate the area under the graph of 25−x 2 from x = 0 to x = 5 using 5 left rectangles.
b. Sketch for yourself the graph of f and the rectangles. Is this left sum an overestimate or underestimate? Enter OVER or UNDER.
c. Now use 5 right rectangles to estimate the area.
d. Is the right sum you got an overestimate or underestimate? Enter OVER or UNDER.
y = 25 - x^2
x _____ y
0 _____ 25
1 _____ 24
2 _____ 21
3 _____ 16
4 _____ 11
delta x = 1
so
25 + 24 + 21 +16 + 11 = 97
b the rectangles are all ABOVE the curve, so the estimate is OVER
c do it now with x = 1, 2, 3, 4, 5
these rectangles are below the curve so under estimate
real answer by the way:
integral from x = 0 to x = 5 of 25-x^2
25 x - (1/3)x^3 at 5 - at 0
125 - (1/3)(125) = (2/3)(125) =83 1/3
To estimate the area under the graph of 25 - x^2 from x = 0 to x = 5 using 5 left rectangles, you would follow these steps:
Step 1: Divide the interval [0, 5] into 5 equal subintervals. Since there are 5 rectangles, each subinterval will have a width of (5 - 0) / 5 = 1.
Step 2: Determine the left endpoint of each subinterval. The left endpoints will be the x-values of the intervals. In this case, the left endpoints are: 0, 1, 2, 3, 4.
Step 3: Evaluate the function at each left endpoint to find the corresponding y-values. In this case, the function is 25 - x^2. So, the y-values at the left endpoints would be: 25, 24, 21, 16, 9.
Step 4: Calculate the area of each rectangle. The area of each rectangle is given by the height (y-value) multiplied by the width (1). So, the areas of the 5 rectangles would be: 25 * 1, 24 * 1, 21 * 1, 16 * 1, 9 * 1.
Step 5: Sum up the areas of all the rectangles to find the estimated area under the graph. Adding up the areas of the 5 rectangles, you would get: (25 * 1) + (24 * 1) + (21 * 1) + (16 * 1) + (9 * 1) = 95.
b. To determine if this left sum is an overestimate or underestimate, you would need to compare it to the actual area under the curve. The left sum includes the area of the rectangles that lie to the left of the curve, but it does not include the area of the rectangles that lie above the curve. In this case, if you sketch the graph of the function and the rectangles, you will see that the left sum is an underestimate of the actual area under the graph. Therefore, the answer is UNDER.
c. To estimate the area using 5 right rectangles, you would follow the same steps as above, but now using the right endpoints of each subinterval.
Step 1: Divide the interval [0, 5] into 5 equal subintervals with a width of 1.
Step 2: Determine the right endpoints of each subinterval. The right endpoints will be the x-values of the intervals. In this case, the right endpoints are: 1, 2, 3, 4, 5.
Step 3: Evaluate the function at each right endpoint to find the corresponding y-values. In this case, the function is 25 - x^2. So, the y-values at the right endpoints would be: 24, 21, 16, 9, 0.
Step 4: Calculate the area of each rectangle. The area of each rectangle is given by the height (y-value) multiplied by the width (1). So, the areas of the 5 rectangles would be: 24 * 1, 21 * 1, 16 * 1, 9 * 1, 0 * 1.
Step 5: Sum up the areas of all the rectangles to find the estimated area under the graph. Adding up the areas of the 5 rectangles, you would get: (24 * 1) + (21 * 1) + (16 * 1) + (9 * 1) + (0 * 1) = 70.
d. Similar to part b, you would need to compare the right sum to the actual area under the curve to determine if it is an overestimate or underestimate. In this case, if you sketch the graph of the function and the rectangles, you will see that the right sum is an overestimate of the actual area under the graph. Therefore, the answer is OVER.