The approximate area between y=3x^(2)+1 and the x-axis on the interval -1 to 5 is 136.6875.
We can approximate the area between y=3x^(2)+1 and the x-axis on the interval -1 to 5 using a Riemann sum with 4 rectangles of equal width. The left-hand endpoint of each subinterval will be used.
The width of each rectangle is the length of the interval (5-(-1)) divided by the number of rectangles (4). Therefore, the width of each rectangle is
(5-(-1))/4 = 6/4 = 1.5.
Since we are using the left-hand endpoint of each subinterval, the x-values of the left-hand endpoints of the rectangles are
-1, -1+1.5
-1+1.5+1.5
-1+1.5+1.5+1.5 = -1+4.5=3.5
The height of each rectangle is determined by the value of y at the corresponding x-value.
Therefore, the heights of the rectangles are
3(-1)^2+1=2
3(-1+1.5)^2+1=7.125
3(3.5)^2+1=35.125
3(3.5+1.5)^2+1=46.875
The area of each rectangle is the product of its width and height, so the areas of the rectangles are
2*1.5=3
7.125*1.5=10.6875
35.125*1.5=52.6875
46.875*1.5=70.3125.
Therefore, the approximate area between y=3x^(2)+1 and the x-axis on the interval -1 to 5 is 136.6875.
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