Area Under A Curve And Writing a Riemann’s Sum Catherine Bernal Period 2

Given the Equation y =x²+1, find the area under the curve from x=1 to x=4 using 5 left hand rectangles. First, graph the equation and draw in your rectangles. Y = x2 +1 y Height of rectangle = f(x) f( ) 3/5 Width of rectangle = (x2 – x1) ÷ Number of rectangles f( ) f( ) 3/5 f( ) 3/5 f(1) 3/5 3/5 2 x So… Width = (4-1) / 5 or 3/5 1 2 3 4

To find the area under the curve, you must add up the areas of each rectangle. y= x2+1 An easier way to show your calculations is by creating a Riemann’s Sum equation. [(Width x heights)= Area] f( ) 3/5 f( ) Area= [f(1)+f( )+f( )+f( )+f( )] f( ) 3/5 f( ) 3/5 f(1) 3/5 3/5 Note: You do not calculate f(4) because we are using Left Hand Rectangles, not right hand. 2 x 1 2 3 4

Writing a Riemann’s Sum Equation… 1. Write Sigma 4. Write in the width of the rectangles after (or before) the sigma. 2. On the bottom of the sigma, you write your starting point. 5. Write parenthesis followed by the equation of the curve. 3. On the top, write in the number you wish to end at (Since we do not calculate f(4), you must calculate the last height before f(4) which in this case would be f( ) 17/5 (x2+1) X=1 You Now Have A Riemann’s Sum Equation!

Area Under a Curve Using Right Hand Rectangles… Now, find the area using the same methods as used with left hand rectangles: 1. The Heights are going to start from the right side this time so the first rectangle drawn is going to start at x = 4 y f( 4 ) y= x2+1 [(Width x heights)= Area] f( ) Area= [f(4) + f( )+f( )+f( )+f( )] 2. Width is the same as when we used Left hand Rectangles because we are still using the same amount of rectangles. f( ) f( ) Now create a Riemann’s sum for your new area calculations… f( ) 3/5 3/5 3/5 3/5 3/5 x Note: You do not calculate the height for f(1) because we are using Right Hand Rectangles. 1 2 3 4

Writing a Riemann’s Sum for Right Hand Rectangles The only differences between writing a Riemann’s Sum for right hand rectangles and left hand rectangles are the starting and ending points. 1. Write your Sigma 4. Write in the width of the rectangles after (or before) the sigma (just as with the left hand notation). 2. Write in your starting point. (since you are using right hand rectangles & do not calculate f(1), your starting point is going to be 3/5 after x=1) 5. Write parenthesis followed by the equation of the curve. 3. On the top, write in your endpoint. 4 (x2+1) X= 8/5

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