In this question, you will estimate the area under the curve y = 4x^2 from x = 2

Question

In this question, you will estimate the area under the curve y = 4x^2 from x = 2 to x = 4 using three different Riemann sums. You will subdivide the interval [2,4] into four sub-intervals of equal width. A. Using our standard notation for Riemann sums, enter the values of a, b, n, and x. a = b = n = x = 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 x1: _____ f (x1):_____ x2: _____ f (x2):_____ x3: _____ f (x3):_____ x4: _____ f (x4):_____ D. Now calculate the approximate area under the curve using the formula Area= {f(x1)+f(x2)+f(x3+f(x4)} *deltax 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 (x1):_____ x2:_____ f (x2):_____ x3:_____ f (x3):_____ x4:_____ f (x4):_____ F. Now calculate the approximate area under the curve using the formula Area= {f(x1)+f(x2)+f(x3+f(x4)} *deltax 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 (x1):____ x2:____ f (x2):____ x3:____ f (x3):____ x4:____ f (x4):____ H. Now calculate the approximate area under the curve using the formula Area= {f(x1)+f(x2)+f(x3+f(x4)} *deltax Approximate area under curve : _______

Explanation / Answer

4x^2 First subinterval: [2, 2.5] Second subinterval: [2.5, 3] Third subinterval: [3, 3.5] Fourth subinterval: [3.5, 4] (Left endpoints) x1: 2 f(x1): 4(2^2) = 4(4) = 16 x2: 2.5 f(x2): 4(2.5^2) = 4(6.25) = 25 x3: 3 f(x3): 4(3^2) = 4(9) = 36 x4: 3.5 f(x3): 4(3.5^2) = 4(12.25) = 49 Area: 16 + 25 + 36 + 49 = 126 (Right end-points) x1: 2.5 f(x1): 25 x2: 3 f(x2): 36 x3: 3.5 f(x3): 49 x4: 4 f(x4): 4(4^2) = 4(16) = 64 Area: 25 + 36 + 49 + 64 = 174 (mid-points) x1: 2.25 f(x1): 20.25 x2: 2.75 f(x2): 30.25 x3: 3.25 f(x3): 42.25 x4: 3.75 f(x4): 56.25 Area: 20.25 + 30.25 + 42.25 + 56.25 = 149

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