Use upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width). y = sqrt 6x

Question

The interval [0,1] is subdivided into 4 subintervals. Round to three decimal places.

Answer 1

Left Right

x f(x) x f(x)

0 0 0.25 1.224744871

0.25 1.224744871 0.5 1.732050808

0.5 1.732050808 0.75 2.121320344

0.75 2.121320344 1 2.449489743

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totals 5.078116023 7.527605765

these totals are multiplied by the step size which is deltaX=0.25 = 1/4

the area are APPROXIMATELY 1.269529 and 1.881901

Now for the EXACT area......

The anti-derivative:

sqrt(6x) dx U=6x

dU = 6 dx

(1/6) dU = dx

integral [ sqrt(6x) dx ] =

(1/6) integral [ sqrt(U) dU ]

(1/6) integral [ U^(1/2) dU ]

(1/6) (2/3) U^(3/2)

(1/9) U ^(3/2)

(1/9) (6x)^(3/2)

lim as x-->0 is zero

lim as x-->1 is (1/9)(6)^(3/2) which is APPROXIMATELY 1.632993.....

The exact area upon subtracting these limits: 1.632993.....

which is between the lower and upper bounds

Answer 2

The upper sum would use the right sides of each subdivision to make rectangles resulting in the area being 0.25•1.225 + 0.25•1.732 + 0.25•2.121 + 0.25•2.449 = 1.882

The lower sum would use the left sides of each subdivision to make rectangles resulting in the area being 0.25•0 + 0.25•1.225 + 0.25•1.732 + 0.25•2.121 = 1.270

The true value is somewhere between these two values so we can take a rough cut and average the two to get: (1.882 + 1.270)/2 = 1.576