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Riemann Sums Jim Wang Mr. Brose Period 6
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Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4 rectangles whose height is given using the right endpoint c)4 rectangles whose height is given using the midpoint d)4 trapezoids e)Evaluate the integral directly
![Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4 rectangles whose height is given using the right endpoint c)4 rectangles whose height is given using the midpoint d)4 trapezoids e)Evaluate the integral directly](http://images.slideplayer.com/15/4748657/slides/slide_2.jpg "Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4 rectangles whose height is given using the right endpoint c)4 rectangles whose height is given using the midpoint d)4 trapezoids e)Evaluate the integral directly")
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a) 4 rectangles whose height is given using the left endpoint Left: ( b h ) (1)0 + (1)1 + (1)4 + (1)9 1 2 3 4 Ans. = 14
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b) 4 rectangles whose height is given using the right endpoint Right: ( b h ) (1)1 + (1)4 + (1)9 + (1)16 1 2 3 4 Ans. = 30
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c) 4 rectangles whose height is given using the midpoint Midpoint: ( b h) (1)1/4 + (1)9/4 + (1)25/4 + (1)49 1 2 3 4 Ans. = 84/4 or 21
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d) 4 trapezoids Area of Trapezoid: ½ h ( b 1 + b 2 ) ½ (1) [ 0+1 + 1+4 + 4+9 + 9+16 ] 1 2 3 4 ½ [ 44 ] Ans. = 22
![d) 4 trapezoids Area of Trapezoid: ½ h ( b 1 + b 2 ) ½ (1) [ ] ½ [ 44 ] Ans.](http://images.slideplayer.com/15/4748657/slides/slide_6.jpg "= 22.")
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e) Evaluate the integral directly Insert in Calculator: f(x) = 0 4 x² dx Ans. = 64/3
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