h. Now calculate the approximate area under the curve using the formula Arm\" {f(f1)+f(~1'2)+f(~1'3)+f(~1'1)}A~1' Approximate area under curve : C] l. You can calculate the exact area under the curve using the Fundamental Theorem of Calculus: 3 Area: / 2I'idr = 17(3) FU) . 1 where F (x) is an antiderivative of the function f (x) = 2x3. Find an antiderivative of the function f (X). Enter your constant of integration as c. Now find the exact value of the area under the curve. Notice that your value of c does not matter when you calculate the area under the cun/e. You should notice that the left end-point calculation under-estimates the area and that the right end-point calculation over-estimates the area. This is because the function is increasing. In this question, you will estimate the area under the curve y = 2x3 from x = 1 to x = 3 using three different Riemann sums. You will subdivide the interval [1,3] into four sub-intervals of equal width. a. Using our standard notation for Riemann sums, enter the values of a, b, n, and Ax. b = n= Ax = b. Complete the following table listing the four sub-intervals: First subinterval : [ Second subinterval : [ Third subinterval : [ Fourth subinterval : [ c. In the next two parts of the question, you will calculate the approximate area under the curve using the left end-points of the sub-intervals. Complete the following table X 1 : f ( x 1 ) : X2 : f ( x 2 ) : [ X3 f ( x 3 ) : x4 : f ( x 4 ) : [ d. Now calculate the approximate area under the curve using the formula Area ~{f (x1) + f (2) + f (x3) + f (x4) }Ax Approximate area under curve : e. In the next two parts of the question, you will calculate the approximate area under the curve using the right end-points of the sub-intervals. Complete the following table (for the second column, you should only have to do one calculation from scratch, since you have already calculated three of the numbers earlier in the question). X1 f ( x 1 ) : X 2 : f ( x 2 ) : X3 : f ( x 3 ) : x4 : F ( x 4 ) : [ f. Now calculate the approximate area under the curve using the formula Area ~{f (x1) + f (x2) + f (x3) + f (x4) }Ax Approximate area under curve : g. In the next two parts of the question, you will calculate the approximate area under the curve using the mid-points of the sub-intervals. Complete the following table X1 : f ( x 1 ) : x 2 : f ( x 2 ) : X3 : f ( x 3 ) : XA : F ( X 4 ) : 1