Estimate the area under the graph of

f(x)=sin(pix) from x=0 to x=1 using the areas of 3 rectangles of equal width, with heights of the rectangles determined by the height of the curve at

a) left endpoint:
b) right endpoint:

what's the trouble?

You have 3 rectangles, each of width 1/3
Using left endpoints, the heights are
0,√3/2,√3/2
add up the rectangle areas.

Similarly, using right endpoints.

I tried that, and it was incorrect

To estimate the area under the graph of a function using rectangles, we can use the method of Riemann sums. In this case, we will divide the interval from x = 0 to x = 1 into three equal subintervals and use the left and right endpoints of each subinterval to determine the heights of the rectangles.

a) Estimating with the left endpoints:
1. Divide the interval [0, 1] into three equal subintervals:
- Subinterval 1: [0, 1/3]
- Subinterval 2: [1/3, 2/3]
- Subinterval 3: [2/3, 1]

2. We will estimate the area under the curve using the left endpoint of each subinterval as the height of the rectangle.

3. Calculate the width of each subinterval:
- The width of each subinterval is 1/3 (since we divided the interval into 3 equal parts).

4. Calculate the height of each rectangle:
- Evaluate the function f(x) = sin(πx) at the left endpoint of each subinterval:
- For subinterval 1: f(0) = sin(π*0) = 0
- For subinterval 2: f(1/3) = sin(π*(1/3)) ≈ 0.866
- For subinterval 3: f(2/3) = sin(π*(2/3)) ≈ 0.866

5. Calculate the area of each rectangle:
- Area of rectangle 1 = width * height = (1/3) * 0 = 0
- Area of rectangle 2 = width * height = (1/3) * 0.866 ≈ 0.289
- Area of rectangle 3 = width * height = (1/3) * 0.866 ≈ 0.289

6. Sum the areas of all three rectangles to get the estimated area under the curve:
- Estimated area = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3
= 0 + 0.289 + 0.289
≈ 0.578

Therefore, the estimated area under the graph of f(x) = sin(πx) from x = 0 to x = 1 using the heights determined by the left endpoints of the subintervals is approximately 0.578 square units.

b) Estimating with the right endpoints:
To estimate the area using the right endpoints, you would follow a similar process as above, but instead, you would use the right endpoint of each subinterval to determine the height of the rectangles. Repeat steps 1-5 using the right endpoints for each subinterval.

For f(x) = sin(πx), evaluating the function at the right endpoints of each subinterval gives:
- For subinterval 1: f(1/3) = sin(π*(1/3)) ≈ 0.866
- For subinterval 2: f(2/3) = sin(π*(2/3)) ≈ 0.866
- For subinterval 3: f(1) = sin(π*1) = 0

Using these heights, follow steps 4-6 again to calculate the estimated area under the curve. You will get a different result than when using the left endpoints.