Presentation on theme: "Area Under A Curve And Writing a Riemanns Sum Catherine Bernal Period 2."— Presentation transcript:
Area Under A Curve And Writing a Riemanns 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 = x 2 +1 y x Height of rectangle = f(x) f(1) 2 Width of rectangle = (x 2 – x 1 ) ÷ Number of rectangles So… Width = (4-1) / 5 or 3/5 3/5 f( ) 3/5
An easier way to show your calculations is by creating a Riemanns Sum equation. To find the area under the curve, you must add up the areas of each rectangle. f(1) 2 3/5 f( ) 3/5 Area= [f(1)+f( )+f( )+f( )+f( )] y x y= x 2 +1 [(Width x heights)= Area] Note: You do not calculate f(4) because we are using Left Hand Rectangles, not right hand.
Writing a Riemanns Sum Equation… 1. Write Sigma 2. On the bottom of the sigma, you write your starting point. X=1 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( ) 4. Write in the width of the rectangles after (or before) the sigma. 5. Write parenthesis followed by the equation of the curve. (x 2 +1) You Now Have A Riemanns Sum Equation! 17/5
Area= [ f(4) + f( )+f( )+f( )+f( )] 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 f( ) Area Under a Curve Using Right Hand Rectangles… f( ) 3/5 f( ) f( 4 ) 3/ y x y= x Width is the same as when we used Left hand Rectangles because we are still using the same amount of rectangles. Note: You do not calculate the height for f(1) because we are using Right Hand Rectangles. Now, find the area using the same methods as used with left hand rectangles: [(Width x heights)= Area] Now create a Riemanns sum for your new area calculations…
Writing a Riemanns Sum for Right Hand Rectangles 1. Write your Sigma 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) X= 8/5 3. On the top, write in your endpoint. 4. Write in the width of the rectangles after (or before) the sigma (just as with the left hand notation). 5. Write parenthesis followed by the equation of the curve. 4 The only differences between writing a Riemanns Sum for right hand rectangles and left hand rectangles are the starting and ending points. (x 2 +1)
You Now Know How To Find the Area Under A Curve Using Rectangles And How To Write A Riemanns Sum!