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 a
a) left endpoint:
b) right endpoint:

A nice calculator for Riemann sums is at

http://mathworld.wolfram.com/RiemannSum.html

To estimate the area under the graph of f(x) = sin(πx) from x = 0 to x = 1 using the areas of 3 rectangles, we need to divide the interval [0, 1] into 3 equal parts.

a) Left endpoint:
First, let's divide the interval [0, 1] into 3 equal parts. The width of each rectangle will be 1/3.

For the left endpoint estimation, we consider the left endpoint of each subinterval as the height of the rectangle.

- For the first rectangle, the left endpoint is x = 0, so the height is f(0) = sin(π*0) = 0.
- For the second rectangle, the left endpoint is x = 1/3, so the height is f(1/3) = sin(π*(1/3)).
- For the third rectangle, the left endpoint is x = 2/3, so the height is f(2/3) = sin(π*(2/3)).

Now, let's calculate the area of each rectangle:

- The area of the first rectangle is (height * width) = (0 * 1/3) = 0.
- The area of the second rectangle is (height * width) = (sin(π*(1/3)) * 1/3).
- The area of the third rectangle is (height * width) = (sin(π*(2/3)) * 1/3).

To get the total estimated area under the graph, we add up the areas of all three rectangles:

Total estimated area = area of first rectangle + area of second rectangle + area of third rectangle.

b) Right endpoint:
For the right endpoint estimation, we consider the right endpoint of each subinterval as the height of the rectangle.

- For the first rectangle, the right endpoint is x = 1/3, so the height is f(1/3) = sin(π*(1/3)).
- For the second rectangle, the right endpoint is x = 2/3, so the height is f(2/3) = sin(π*(2/3)).
- For the third rectangle, the right endpoint is x = 1, so the height is f(1) = sin(π*1) = 0.

Again, let's calculate the area of each rectangle:

- The area of the first rectangle is (height * width) = (sin(π*(1/3)) * 1/3).
- The area of the second rectangle is (height * width) = (sin(π*(2/3)) * 1/3).
- The area of the third rectangle is (height * width) = (0 * 1/3) = 0.

To get the total estimated area under the graph, we add up the areas of all three rectangles:

Total estimated area = area of first rectangle + area of second rectangle + area of third rectangle.

To estimate the area under the graph of f(x) = sin(πx) using rectangles, we need to split the interval from x=0 to x=1 into multiple equal-width subintervals. In this case, we will use 3 rectangles of equal width.

The width of each rectangle will be (1 - 0)/3 = 1/3.

a) Left Endpoint Approximation:
For the left endpoint approximation, we consider the left endpoint of each subinterval to determine the height of the rectangle.

Since our subintervals have equal widths of 1/3, the left endpoints of the intervals are 0, 1/3, 2/3. We evaluate f(x) = sin(πx) at these left endpoints to determine the height of each rectangle.

Height of the first rectangle: f(0) = sin(π(0)) = 0
Height of the second rectangle: f(1/3) = sin(π(1/3)) ≈ 0.866
Height of the third rectangle: f(2/3) = sin(π(2/3)) ≈ 0.866

Now, we calculate the area of each rectangle using the height and width of the rectangle:

Area of the first rectangle: (1/3) * 0 = 0
Area of the second rectangle: (1/3) * 0.866 ≈ 0.289
Area of the third rectangle: (1/3) * 0.866 ≈ 0.289

To estimate the total area under the graph, we sum up the areas of all the rectangles:

Total area ≈ 0 + 0.289 + 0.289 = 0.578

Therefore, using the left endpoint approximation, the estimated area under the graph of f(x) = sin(πx) from x=0 to x=1 is approximately 0.578.

b) Right Endpoint Approximation:
For the right endpoint approximation, we consider the right endpoint of each subinterval to determine the height of the rectangle.

Since our subintervals have equal widths of 1/3, the right endpoints of the intervals are 1/3, 2/3, 1. We evaluate f(x) = sin(πx) at these right endpoints to determine the height of each rectangle.

Height of the first rectangle: f(1/3) = sin(π(1/3)) ≈ 0.866
Height of the second rectangle: f(2/3) = sin(π(2/3)) ≈ 0.866
Height of the third rectangle: f(1) = sin(π(1)) = 0

Now, we calculate the area of each rectangle using the height and width of the rectangle:

Area of the first rectangle: (1/3) * 0.866 ≈ 0.289
Area of the second rectangle: (1/3) * 0.866 ≈ 0.289
Area of the third rectangle: (1/3) * 0 = 0

To estimate the total area under the graph, we sum up the areas of all the rectangles:

Total area ≈ 0.289 + 0.289 + 0 = 0.578

Therefore, using the right endpoint approximation, the estimated area under the graph of f(x) = sin(πx) from x=0 to x=1 is approximately 0.578.