Estimate the area under the graph of f(x)= x^2 + 3 x from x=1 to x=10 using the areas of 3 rectangles of equal width, with heights of the rectangles determined by the height of the curve at

a) left endpoints:
b) right endpoints:

so the values of x would be

x = 1,4,7, and 10 , equal width = 3

left end-points
x = 1 , 4, 7
f(1) = 1+3 = 4
f(4) = 16+12 = 28
f(7) = 49+21 = 70
area = 3(4) + 3(28) + 3(70) = 306 ---> under-estimation

right end-points
x = 4 , 7, and 10
f(4) = 28
f(7) = 70
f(10) = 130
area = 3(28+70+130) = 684 ---> over-estimation

let's average them:
(306+684)/2 = 495

which is close to what if we had taken the midpoint of the base of each rectangle
f(2.5) = 13.75
f(5.5) = 46.75
f(8.5) = 97.75
area = 3(13.75+46.75+97.75) = 474.75

exact answer by Calculus:
481.5

To estimate the area under the graph of the function f(x) = x^2 + 3x from x = 1 to x = 10 using rectangles, we can use the left and right endpoints of each rectangle to determine the height.

a) Using left endpoints:
To estimate the area using left endpoints, we divide the x-axis interval into 3 equal subintervals since we have 3 rectangles. Each rectangle's height is determined by the value of the function at the left endpoint of that subinterval.

1. Calculate the width of each subinterval by dividing the difference between the right and left endpoints by the number of rectangles (3 in this case):
Width = (10 - 1) / 3 = 3

2. Calculate the left endpoint of each subinterval:
Left endpoint of first subinterval: x = 1
Left endpoint of second subinterval: x = 1 + width = 4
Left endpoint of third subinterval: x = 1 + 2 * width = 7

3. Calculate the height of each rectangle by evaluating the function at each left endpoint:
Height of first rectangle: f(1) = (1^2) + (3 * 1) = 4
Height of second rectangle: f(4) = (4^2) + (3 * 4) = 28
Height of third rectangle: f(7) = (7^2) + (3 * 7) = 70

4. Calculate the area of each rectangle:
Area of first rectangle: height * width = 4 * 3 = 12
Area of second rectangle: 28 * 3 = 84
Area of third rectangle: 70 * 3 = 210

5. Estimate the total area under the graph by summing up the areas of all the rectangles:
Total area ≈ 12 + 84 + 210 = 306

Therefore, using left endpoints, the estimated area under the graph of f(x) = x^2 + 3x from x = 1 to x = 10 is approximately 306 square units.

b) Using right endpoints:
The steps for estimating the area using right endpoints are similar to those we followed for left endpoints. The only difference is that we use the right endpoint of each subinterval instead of the left endpoint.

1. Calculate the width of each subinterval (same as in part a):
Width = (10 - 1) / 3 = 3

2. Calculate the right endpoint of each subinterval:
Right endpoint of first subinterval: x = 1 + width = 4
Right endpoint of second subinterval: x = 4 + width = 7
Right endpoint of third subinterval: x = 7 + width = 10

3. Calculate the height of each rectangle by evaluating the function at each right endpoint:
Height of first rectangle: f(4) = (4^2) + (3 * 4) = 28
Height of second rectangle: f(7) = (7^2) + (3 * 7) = 70
Height of third rectangle: f(10) = (10^2) + (3 * 10) = 130

4. Calculate the area of each rectangle:
Area of first rectangle: height * width = 28 * 3 = 84
Area of second rectangle: 70 * 3 = 210
Area of third rectangle: 130 * 3 = 390

5. Estimate the total area under the graph by summing up the areas of all the rectangles:
Total area ≈ 84 + 210 + 390 = 684

Therefore, using right endpoints, the estimated area under the graph of f(x) = x^2 + 3x from x = 1 to x = 10 is approximately 684 square units.